Question

E and f are mid point of sides and a of a triangle. Bf and Ce are produced to x and y so that ex=ce and fy =bf prove that a x y is straight line

Answer 1

In triangles AEX & BEC, we have AE = BE (since E is the mid-point of AB) angle AEX = angle BEC (V.O.A) EX = EC (given) therefore, angle AEX congruent to angle BEX (SAS) => angle XAE = angle CBE ( by CPCT) => angle XAB = angle CBA (since angle XAE = angle XAB & angle CBE = angle CBA) But, angle XAB & angle CBA are alternate interior angles formed when transversal AB meets XA at A and BC at B. => XA II BC ... (1) Similarly it can be proved that, triangle AFY congruent to triangle CFB & AY II BC ...(2) From (1) & (2),we get BC II XA & BC II AY So, XA II AY Hence, XAY is a straight line