Two circle with centres O and O' touch each other externally at point p and XY is a common tangent to the two circles at P. prove that O,P,O' are collinear.

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Solution

We have

Then,

$O$ and $O^{'}$ be the centre of two circles.

$X Y$ be a common tangent.

P is the mid point of $O a n d O^{'}$ .

Construct:-draw $O X$ and $O X^{'}$ .

Then,

$I n Δ O P X a n d O^{'} P X$

$O P = O^{'} P (r a d i u s o f c i r c l e)$

$P X = P X (C o m m o n l i n e)$

$∠ X P O = ∠ X P O^{'} (E v e r y 90^{o})$

Then,

$Δ O P X ≅ Δ O^{'} P X$
So, $O P O^{'}$ are collinear.

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Two circle with centres O and O' touch each other externally at point p and XY is a common tangent to the two circles at P. prove that O,P,O' are collinear.

We have Then, O and O′be the centre of two circles. XY be a common tangent. P is the mid point of OandO′. Construct:-draw OX and OX′. Then, InΔOPXandO′PX OP=O′P(radiusofcircle) PX=PX(Commonline) ∠XPO=∠XPO′(Every90o) Then, ΔOPX≅ΔO′PX So, OPO′ are collinear.