Theorem 6.9: If triangles are similar then the ratios the lengths of their corresponding sides are all the same.

Proof: We show that if the angles are congruent that the ratios of the lengths of the corresponding sides are all the same.

In the illustration

Since   DA,   we can move   D   to be on top of   A,  by the definition of congruent. If   E   is not on   AB,   (or it's extension) then rotate   ΔDEF   about the angle bisector of   / D,   and we will be able to assume that it is. Then rotate the figure about   A   until   BC   is vertical.

Since AB and AC intersect BC , they are not vertical, and will, thus, have finite real slopes with which we can do arithmetic.

Since / AEF/ B,   EF || BC,   by Theorem 6.2, so   |EF|   is also vertical.

Let us define coordinates for these points. Let

A = D = (xo, yo)
B = (x1, y1)
C = (x2, y2)
E = (x3, y3)
F = (x4, y4)

Since   BC   is vertical,   x1 = x2,   and, since   EF   is vertical,   x3 = x4.   Let

m1   be the slope of   AB

and

m2 be the slope of   AC

Then by Theorem 1.5,


so

and

But, since   x1 = x2,   and   x3 = x4   This gives us

As required. Since   A   and   D   were an arbitrary pair of congruent angles, the same proof could be repeated for the other pairs of corresponding sides.

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