# Midpoint of square's diagonal - proof

• Mar 15th 2011, 11:31 PM
mathguy80
Midpoint of square's diagonal - proof
Hi All,

I am not able to solve this geometry problem correctly. Visually it looks plausible but how do you go about proving it.

Attachment 21159

In the diagram, ABCD is a square, EFG is an isosceles triangle, Angle EFG = 90, and M is midpoint of EG.

(a) Show that Angle EFA = Angle FGB.

(b) Explain why Triangle AFE, and BGF are congruent.

(c) If AC is drawn, explain why M is the midpoint of AC.

Also one more question! Is there a way to describe such geometry problems in latex? I did the the attachment in mspaint, was wondering if there is a better way? Thanks for your help!
• Mar 16th 2011, 03:05 AM
HallsofIvy
The crucial point is that right angle, EFG. Because of that the two angles AFE and GFB are "complementary"- the measures of the three angles, AFE, EFG, and GFB, add up to 180 degrees because they form a straight line and since EFG is a right angle, AFE and GFB add to 90 degrees.

And then, look at right triangles EAF and FGB. Angle EFA is complementary to AEF and FGB is complementary to GFB. That, together with the fact that AFE and GFB are complementary, shows that EFA and FGB are congruent.

For (b) you need the additional given information, that "EFG is an isosceles triangle". That is, sides EF and FG are congurent. Then use "ASA".

For (c), notice that drawing horizontal and vertical lines through M divides the square into 4 smaller but identical squares.
• Mar 16th 2011, 07:33 AM
mathguy80
Thanks @HallsofIvy. That clears up the fog quite a bit!

Regarding (a) is this a corollary like ASA, SSS, etc of similar/rt triangles, ie:- if non right angles of right triangles are complementary it is similar?

Regarding (c) can you please clarify how to show that the points A, C and M are collinear?

Thanks a lot!
• Mar 16th 2011, 10:29 AM
emakarov
Quote:

Also one more question! Is there a way to describe such geometry problems in latex? I did the the attachment in mspaint, was wondering if there is a better way?
Check out this FAQ. On this forum, you can use the picture environment. See an example in this thread. Elsewhere, PGF/TikZ is a very powerful graphic language for TeX. Its author also wrote an awesome Beamer class for creating presentations. However, the learning curve for TikZ is pretty steep.
• Mar 17th 2011, 06:54 AM
mathguy80
Thanks @emakarav. I'll check it out.