Question

PROVE THE FOLLOWING RESULT: Using a compass and a straightedge, construct a triangle ABC such that side BC is of length 2, the circumradius is of length 3/2 and the median AA' is of length 2 and is the midpoint of BC.

Answer 1

We can use the definitions of median and circumradius to construct the given triangle.

First we draw the side BC and mark its midpoint A'. As the median from A to A' has length 2, all points lying at distance 2 from A' are suitable candidates for the third vertex. These are just the points on the circle of radius 2 with center A'.

Then, we use the circumradius. If we draw a circle with radius 3/2 and let it be the circumcircle for our triangle, BC is a chord of length 2 for this circle. In addition, the point A also has to lie on this circle.

So, the third vertex has to be the points lying on both these circles, that is, the point of intersection of the 2 circles.

After drawing the triangle, we can easily measure and verify that AA' has the required length.