Problem 4

Construct a triangle

ABC

given the lengths of the altitudes from

A

and

B

and the length of the median from

A

Step 1 of 5: Build the determined right triangle AHM

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In plain words

The altitude and median from A already determine a right triangle around the midpoint of the unknown base.

AH=h_a,\quad AM=m_a,\quad AH\perp HM

Detailed analysis

Let $M$ be the midpoint of $BC$ and $H$ the foot of the altitude from $A$ . The given data prescribe $AH=h_a$ , $AM=m_a$ , and $AH\perp HM$ ; construct this right triangle. The line $HM$ is now the line containing $BC$ .