Proposition III.7 — If on the diameter of a circle a point be taken which is not…

Proof

Let

A D

be a diameter of circle

A B C D

with centre

E

, and let

F

A D

be distinct from

E

. From

F

draw lines

F B

F C

to the circumference. Join

E B

E C

. In

△ E B F

E B + E F > F B

(I.20). But

E B = E A

(radii) and

E F + E A = F A

, so

F A = E F + E B > F B

; hence the line

F A

along the diameter towards the centre is longer than any other. The line

F D

on the other side is similarly the shortest. For intermediate lines

F B

F C

with

B

closer to

A

than

C

, the SAS inequality I.24 in the radius-line-radius triangles gives

F B > F C

when

∠ B E F > ∠ C E F

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Depends on (4)

I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal straight lines be produced…
I.20Proposition I.20In any triangle two sides taken together in any manner are greater than the remaining one.
I.24Proposition I.24If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the…
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

Required by (dependents) (1)

III.8Proposition III.8If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is…

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