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 5 of 5: Conclude that the construction satisfies all three data

Previous step Next step

AH=h_a,\quad AM=m_a,\quad BI=h_b

Detailed analysis

The first construction gives the required altitude $AH=h_a$ and median $AM=m_a$ , while Step 4 proves that the altitude from $B$ is $BI=h_b$ . Therefore the constructed triangle has exactly the three prescribed lengths. Different valid intersections $X$ give the corresponding possible configurations.