a circle touches the sides PQ , QR and PR of triangle PQR at the points X, Y and Z respectively. Show that PX + QY + RZ = XQ + YR + ZP = 1/2 (perimeter of triangle PQR)

Relevance

i) As the circle touches PQ at X and PR at Z; PX = PZ ------------ (1)

[Tangents from the same exterior point to the same circle are equal in measure]

Similarly QY = QX ----------- (2) and

RZ = RY ------------ (3)

ii) Adding (1), (2) & (3),

PX + QY + RZ = PZ + QX + RY ---------------(4) [Proved]

iii) By geometrical addition, PX + XQ = PQ

QY + YR = QR and RZ + ZP = RP

Adding all these 3, PX + XQ + QY + YR + RZ + ZP = PQ + QR + RP

Grouping them conveniently,

(PX + QY + RZ) + (XQ + YR + ZP) = PQ + QR + RP = Perimeter of the triangle ------ (5)

Hence from (4) & (5),

2(PX + QY + RZ) = Perimeter of the triangle

==> PX + QY + RZ = (1/2)(Perimeter of the triangle PQR) ----------- (6)

Thus from (4) & (6),

PX +QY + RZ = XQ + YR + ZP = (1/2)(Perimeter of triangle PQR) [Proved]

• Draw the radius from the circle to X, Y, Z. We know this is perpendicular because the radius of a circle is perpendicular to its tangent. Connect a line from the center of the circle to the vertices. By hypotenuse leg congruency PXC ~ PZC, RZC ~ RYC, QYC ~ QXC. This means PX=XQ, QY=YR, RZ=ZP. This means PX+QY+RZ=XQ+YR+ZP and since those line segments make up the perimeter and are equal PX+QY+RZ=XQ+YR+ZP=1/2 Perimeter (PQR)