Help with Constructing a Triangle

[cyberspace]

Here's the problem:
From the midpoint of one side of a traingle ,segments are drawn perpendicular to the other two sides. Prove that if the segments are congruent, the triangle is isosceles.

Here's the picture that I think represents the statement:

*Note: Triangle is not drawn to scale.

I'm okay with the proof that I have to do, but I'm not sure if my triangle represents the statement properly. Can you guys help me with constructing the correct triangle?

Thanks for your help in advance.

 

[Loren]

Your diagram is not correct. From R draw a line segment perpendicular to BC. That forms a right angle. From R draw another segment perpendicular to AC. That forms a right angle. Along with the given that those two segments are equal, you should be able to identify two right triangles within the original triangle that are right triangles and have two corresponding sides equal.

 

[soroban]

Hello, cyberspace!

From the midpoint of one side of a triangle,
segments are drawn perpendicular to the other two sides.
Prove that if the segments are congruent, the triangle is isosceles.

Loren is right . . . Your diagram should look like this:

              C
              o
             * *
            *   *
           *     *
          *       *
       D o         o E
        * *       * *
       *    *   *    *
    A o * * * o * * * o B
              M

We have \(\displaystyle \Delta ABC.\)

\(\displaystyle M\) is the midpoint of side \(\displaystyle AB\).

Draw \(\displaystyle MD \perp AC\) and \(\displaystyle ME \perp BC.\)

Given that \(\displaystyle MD = ME\), prove that \(\displaystyle AC = BC.\)