Isosceles Triangles

Date: 2/8/96 at 9:47:26
From: Mrs. Doris E. Risenhoover
Subject: Isosceles triangle

I need to find the answer to the following problem:

1.  an isosceles triangle whose points are A, B, & C.
2.  One long side of the triangle has point p
3.  The other side of the trangle has point q
4.  The points p and q are located so that AC=AP=PQ=QB
5.  I need to find angle B, and how to do problem

Date: 3/4/96 at 21:37:9
From: Doctor Elise
Subject: Re: Isosceles triangle

Hi!

Okay, you have an isosceles triangle ABC.  And somewhere on
the sides, you have points Q and P.  I have to assume here
that AP and AQ both already lie on the sides of the triangle--
that is, you don't have point p on the line BC so that you
have to draw AP across the middle of the triangle.  So
we draw an isosceles triangle ABC, and then we put P between A and
B, and Q between A and C, and then we connect P and Q.

Now we look at requirement 4, which says that AC = AP = PQ.
This tells us that the triangle formed by APQ is equilateral, right?
All the angles have to be 60 degrees in APQ, so we know that angle A
is 60 degrees.

Now we have isosceles triangle ABC, with angle A = 60.  If A turns
out to be one of the two equal angles in the isosceles triangle, then
either C or B has to be the same, leaving the third angle to be 60 
degrees also to make up 180 total.  So all three angles would be 60 
degrees.

If angle A is not one of the two equal angles, then we subtract angle A
from 180 degrees and divide the result by 2 to get B and C, and all
three angles STILL end up being 60 degrees.  ABC is an equilateral 
triangle!

I hope this helps.  

P.S.

I can draw an isosceles triangle, where P is between C and D, that meets
all 5 of the requirements, but I can't solve for angle B.  The solution
above doesn't use the clue AP = QB, which makes me a little suspicious 
of my assumption.  Any ideas?

-Doctor Elise,  The Math Forum