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 4 of 5: Use the midpoint theorem to verify the altitude from B

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MX\parallel BI,\quad M\text{ midpoint of }BC\Rightarrow X\text{ midpoint of }CI

Detailed analysis

Because $C,X,A$ are collinear and $\angle AXM=90^\circ$ , the segment $MX$ is perpendicular to $AC$ . Thus $MX$ is parallel to the altitude $BI$ from $B$ to $AC$ . In triangle $BCI$ , $M$ is the midpoint of $BC$ ; the line through $M$ parallel to $BI$ meets $CI$ at its midpoint $X$ . Hence $BI=2MX=h_b$ .