35. The common tangents AB and CD to two circles with centres O and O' intersectalßbetween their centres. Prove that the points O, E and O' are collinearNCERT EXEMPLARİ

35. The common tangents AB and CD to two circles with centres O and O'

1.	35. The common tangents AB and CD to two circles with centres O and O' intersectalßbetween their centres. Prove that the points O, E and O' are collinearNCERT EXEMPLARİ
Answer» ∠AEC =∠DEB (Vertically Opposite Angle)join OA and OCso in triangle OAE and triangle OCE we have,OA=OC (radii of same circle)OE=OE (common)∠OAE=∠OCE (90° as the tangent is always perpendicular to the radius at the point of contact)∴ΔOAE ≡ΔOCEso∠AEO =∠CEO (CPCT)similarly for other circle we have∠DEO' =∠BEO' (CPCT)Now∠AEC=∠DEB⇒1/2(∠AEC) = 1/2(∠DEB)⇒∠AEO =∠CEO =∠DEO'=∠BEO'so all 4∠'s are equal and bisected by OE and OE'∴ O,E,O' are collinear