Show that the perpendicular bisectors of the sides of a triangle are concurrent.
To prove:
Perpendicular bisectors of the sides of a triangle are concurrent.
Assuming:
ABC be a triangle with vertices A (x1, y1), B (x2, y2) and C (x3, y3).
Let D, E and F be the midpoints of the sides BC, CA and AB, respectively.
Explanation:
Thus, the coordinates of D, E and F are
Let mD, mE and mF be the slopes of AD, BE and CF respectively.
∴ Slope of BC × mD = -1
⇒
⇒ mD
Thus, the equation of AD
⇒
⇒
⇒
Similarly, the respective equations of BE and CF are
Let L1, L2 and L3 represent the lines (1), (2) and (3), respectively.
Adding all the three lines,
We observe:
1 ⋅ L1 + 1⋅L2 + 1⋅L3 = 0
Hence proved, the perpendicular bisectors of the sides of a triangle are concurrent.