ABC is a triangle with midpoints for its sides P,Q, and R.
Let l and m denote the
perpendicular lines at P and Q.
If l and m are parallel, then AB and BC would also be parallel, but clearly , AB and BC are not parallel lines, so l and m are not parallel.
Suppose l and m intersect at O.
Then Tri (APO) is congruent to Tri(BPO) [SAS] and so
AO is congruent to BO. [CPCTC]
Similarly BO is congruent to CO.
Now we have that Tri(AOR) is congruent to Tri(COR) ... [SSS],
so <ARO is congruent to <CRO... [CPCTC] .
But these angles are supplementary, so they are right angles.
Thus OR is the perpendicular bisector of AC and the three pendicular bisectors of Tri (ABC) all pass through the point O.
fact O is the center of the circle passing through the three vertices of
the triangle A,B, and C. O is the center of the circumscribed circle -
called the circumcenter- and the sides of the triangle are all chords of