Two circles touch externally at a point P and a common tangent touches them at A and B Prove that - Maths - Circles

Question

Two circles touch externally at a point P and a common tangent
touches them at A and B Prove that AB subtends a right angle at
P

Ajanta Trivedi · Accepted Answer

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/discuss_editlive/4123606/2012_12_28_18_29_48/twocirclestouchexternally_8630650296551779556%20jpg">
given: C1 and C2 are two circles touch each other externally at
P AB is the common tangent to the circles C1 and C2 at point A and
B respectively
TPT:âˆ APB= 90 deg
Proof: let âˆ CAP=x and
âˆ CBP=y
CA=CP [lengths of the tangents from an external point C]
therefore in triangle PAC,
âˆ CAP=âˆ APC=x
similarly CB=CP and
âˆ CPB=âˆ PBC=y
now in the triangle APB,

âˆ PAB+âˆ PBA+âˆ APB=180
[sum of the interior angles in a triangle]
x+y+(x+y)=180
2x+2y=180
x+y=90
therefore âˆ APB=x+y=90
hence AB subtends a right angle at P

Two circles touch externally at a point P and a common tangent touches them at A and B.Prove that AB subtends a right angle at P.