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Amusements in Mathematics
Amusements in Mathematics

Chapter XVII

Cover illustration for Amusements in Mathematics
Context Illustration Reading 1/83 · 5 lines
English subtitles · tap any word to listen 1 / 5
Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.
Also mark off the point G so that DG shall equal DC.
Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.
A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.
The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.
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AI-generated narration by OpenAI · Slow US English
LEARNER DICTIONARY
Archivist
/ˈɑːrkɪvɪst/
Editorial verification pending

A keeper or guardian of historical records and memories.

Amusements in Mathematics

Chapter XVII

Reading 1/83 · 5 reading units · Lines 1–5

Line 1
Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.
Line 2
Also mark off the point G so that DG shall equal DC.
Line 3
Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.
Line 4
A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.
Line 5
The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

Learner Dictionary

Oxford-style entry order: headword, part of speech, US IPA, concise definition, contextual example, audio, and collocations from this reading.

Although noun US /ˈalthough/

a story-specific word describing an important person, object, action, or idea in this passage

From this reading “The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.”

Being auxiliary verb US /ˈbiːɪŋ/

the -ing form of “be”

From this reading “Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.”

Have auxiliary verb US /hæv/

used with a past participle to form the perfect tenses

From this reading “The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.”

Make noun US /make/

an important story-specific word used in this passage

From this reading “Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.”

May modal verb US /meɪ/

used to say something is possible, or to give permission

From this reading “The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.”

Must modal verb US /mʌst/

used to say something is necessary, or almost certainly true

From this reading “Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.”

Partly noun US /partly/

an important story-specific word used in this passage

From this reading “The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.”

Point noun US /pɔɪnt/

the main matter or purpose being discussed

From this reading “Also mark off the point G so that DG shall equal DC.”

Shall modal verb US /ʃæl/

used with I and we for the future, or to offer and suggest

From this reading “Also mark off the point G so that DG shall equal DC.”

Until preposition US /ənˈtɪl/

up to a particular time

From this reading “Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.”

All words in this reading · 91

Every word is available to listen to. Dictionary details are shown only when verified; these are not licensed Oxford entries.

adeterminerUS /ə/

used before a singular noun to mean one of a group, when it is not a particular one

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

all

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

Also

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

althoughnounUS /ˈalthough/

a story-specific word describing an important person, object, action, or idea in this passage

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

andconjunctionUS /ənd/

used to join words or parts of a sentence that go together

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

any

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

asconjunctionUS /æz/

at the same time that; because; in the way that

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

BC

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

BCF

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

BEauxiliary verbUS /biː/

used with a past participle to form the passive, or with -ing for continuous tenses

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

beingauxiliary verbUS /ˈbiːɪŋ/

the -ing form of “be”

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

BF

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

byprepositionUS /baɪ/

showing who or what does the action

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

card

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

cardboard

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

CD

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

CF

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

CG

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

connection

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

corner

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

curious

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

cut

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

cuts

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

D

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

DC

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

DG

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

dotted

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

draw

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

equal

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

equilateral

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

fact

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

Fix

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

forprepositionUS /fɔːr/

intended to be given to, or used by, somebody

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

fromprepositionUS /frʌm/

showing where something starts, or who sent it

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

G

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

given

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

H

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

HA

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

haveauxiliary verbUS /hæv/

used with a past participle to form the perfect tenses

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

IpronounUS /aɪ/

the person speaking or writing, as the subject of the verb

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

IfconjunctionUS /ɪf/

on condition that; in the event that

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

inprepositionUS /ɪn/

inside something, or within a place, period, or state

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

independent

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

indicated

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

isauxiliary verbUS /ɪz/

the form of “be” used with he, she, it, or a singular noun

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

itpronounUS /ɪt/

a thing, animal, or idea already mentioned

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

line

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

made

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

makenounUS /make/

an important story-specific word used in this passage

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

mark

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

maymodal verbUS /meɪ/

used to say something is possible, or to give permission

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

method

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

more

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

mustmodal verbUS /mʌst/

used to say something is necessary, or almost certainly true

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

now

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

obvious

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

ofprepositionUS /ʌv/

belonging to, or connected with, somebody or something

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

off

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Also mark off the point G so that DG shall equal DC.

onprepositionUS /ɑːn/

touching and supported by a surface

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

other

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

ourdeterminerUS /ˈaʊər/

belonging to the speaker and others

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

paper

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

parallel

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

partly

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

piece

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

pointnounUS /pɔɪnt/

the main matter or purpose being discussed

From this reading Also mark off the point G so that DG shall equal DC.

position

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

preferred

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

problem

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

produce

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

reason

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

seen

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

shallmodal verbUS /ʃæl/

used with I and we for the future, or to offer and suggest

From this reading Also mark off the point G so that DG shall equal DC.

side

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

soconjunctionUS /soʊ/

for that reason; with the result that

From this reading Also mark off the point G so that DG shall equal DC.

solution

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

solved

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

thanconjunctionUS /ðæn/

used to introduce the second part of a comparison

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

thatconjunctionUS /ðæt/

used to introduce what somebody said, thought, or felt

From this reading Also mark off the point G so that DG shall equal DC.

thedeterminerUS /ðə/

used before a noun that is already known, already mentioned, or the only one of its kind

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

then

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

thisdeterminerUS /ðɪs/

used for a person or thing that is near, or that has just been mentioned

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

toparticleUS /tuː/

used before the base form of a verb to make the infinitive

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

triangle

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.

untilprepositionUS /ənˈtɪl/

up to a particular time

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

wepronounUS /wiː/

the speaker together with one or more other people

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

whichpronounUS /wɪtʃ/

used to introduce a clause giving more about a thing just mentioned

From this reading Draw the line CG and produce it until it cuts the line BF in H. If we now make HA parallel to BE, then A is the point from which our cut must be made to the corner D, as indicated by the dotted line.

why

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

withprepositionUS /wɪð/

in the company of; together with

From this reading A curious point in connection with this problem is the fact that the position of the point A is independent of the side CD.

working

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading The reason for this is more obvious in the solution I have given than in any other method that I have seen, and (although the problem may be solved with all the working on the cardboard) that is partly why I have preferred it.

yourdeterminerUS /jɔːr/

belonging to the person or people being spoken to

From this reading Fix your card on a piece of paper and draw the equilateral triangle BCF, BF and CF being equal to BC.