Q5 of 70 Page 131

In Δ ABC, the bisector of ∠ A intersects BC in D. The bisector of ∠ADB intersects AB in F and the bisector of ∠ ADC intersects in E. Prove that AF X AB X CE = AE × AC × BF.


Given : In Δ ABC, the bisector of ∠ A intersects in D. The bisector of ∠ADB intersects in F and the bisector of ∠ ADC intersects in E


To prove : AF × AB × CE = AE × AC × BF


Proof : in Δ ABC, the bisector of ∠A intersects at D


∴ 1


(∵ in a Δ the bisector of an angle divides the side opposite to the angle in the segments whose lengths are in the ratio of their corresponding sides)


Similarly, In Δ ADB, the bisector of ∠D intersects at F


∴ ………………2


And in Δ ADC , the bisector of ∠D intersects at E


∴ …………..eq(3)


Multiplying 1, 2 and 3, we get




And
(∵ )


∴


∴ AF × BD × CE =AE × CD × BF


More from this chapter

All 70 →
3

In □ABCD, M and N are the mid-points of and . If ∥ prove that ∥

 4

In □ ABCD, A—P—D, B—Q—C. If AB ∥ PQ and PQ || DC, prove that

AP X QC = PD X BQ.


 6

In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity. Prove that ABC ↔ XYZ is similarity.

 7

State giving reasons, whether the following statements are true or false: In all the following questions the line does not contain a side of the triangle.

(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.


(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.


(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.


(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.


(5) In the plane of Δ ABC, a line / can be drawn such that = {P}, = {Q} and = ϕ.