MathLabs

Problem 4

Construct a triangle ABCABC given the lengths of the altitudes from AA and BB and the length of the median from AA.
Step 2 of 5: Choose X on the circle with diameter AM
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X∈{Y:∠AYM=90∘},MX=hb2X\in\{Y:\angle AYM=90^\circ\},\quad MX=\dfrac{h_b}{2}
Detailed analysis

Draw the circle with diameter AMAM. Choose an intersection XX of this circle with the circle centered at MM of radius hb/2h_b/2. Then ∠AXM=90∘\angle AXM=90^\circ by Thales' theorem and MX=hb/2MX=h_b/2 by construction.