Problem
GEO-B2-M09-P013 Median or Altitude
#13
★★★☆☆ Level 3 of 5
Let \(A(-1,0)\), \(B(1,0)\), \(C(u,v)\), where \(v\ne 0\), and let \(M\) be the midpoint of \(AB\). Prove that \(CM\perp AB\) if and only if \(CA=CB\).
The condition \(CM\perp AB\) means that \(CM\) is vertical. Check \(CA=CB\) using squared lengths.
Point \(M=(0,0)\). Since \(AB\) is horizontal, \(CM\perp AB\) if and only if \(u=0\). On the other hand, \(CA^2=(u+1)^2+v^2\), \(CB^2=(u-1)^2+v^2\). Equality of these squares is also equivalent to \(u=0\). Hence the conditions are equivalent.
A good problem on an “if and only if” proof.