An interesting geometry problem

Question

In a ΔMAP, on sides MA and AP, squares are drawn. If P and D are on the same side of AM; and M,E lie on opposite sides of AP. D and E are the centres of the squares on MA and AP respectively. Find the angle between MP and DE.

I have been trying to solve this problem since long time and I've been unable to do so. I have already found an approach using complex numbers but I want a pure geometry solution. The diagram was really complicated and visualising constructions were much harder. Would someone please help me to solve this question? Thanks for help .

Welcome to Math Stack Exchange. Write your questions clearly. Try to avoid to set pictures as your questions. — nmasanta, Oct 31 '19 at 11:30
I think there isn't enough given, anything else about triangle MAP ? — Anas Khaled, Oct 31 '19 at 12:35
@MichalAdamaszek Actually, the two triangles are similar. Homotheticity is a slightly stronger property. — Jaap Scherphuis, Oct 31 '19 at 12:54
@JaapScherphuis Thanks, you are right. And that is the next hint: figuring out which rotation makes those similar triangles into homothetic ones solves the problem ;) — Michal Adamaszek, Oct 31 '19 at 13:25

Quanto · Accepted Answer · 2019-10-31T21:27:21.177

3

Note 45 + ∠EAD + 45 = ∠PAM + 90 $\implies$ ∠EAD = ∠PAM. Then, along with AM=$\sqrt2$ AD and AP=$\sqrt2$ AE, the triangle ADE and AMP are similar.

Since the sides AD and AE are at the 45-degrees with respect to AM and AP, respectively, the third side DE is also at the 45-degrees with respect to MP.

edited Oct 31 '19 at 21:27

answered Oct 31 '19 at 14:11

Quanto

97,352

How does one get that $AM=\sqrt2AD$ and $AP=\sqrt2AE$? – YiFan Tey Oct 31 '19 at 23:53
@YiFan - AD and AE are half diagonals of the two squares. – Quanto Nov 01 '19 at 00:50
thanks, I don't know how I didn't see that.. – YiFan Tey Nov 01 '19 at 03:26
Thank you very much – Harsh Vora Nov 01 '19 at 06:37

An interesting geometry problem

1 Answers1