Answer to Question #142441 in Geometry for Peace College

Question

Let AK, BL, CN be angle bisectors of triangle ABC, and they intersect at point I. It is known that the ratios of areas of triangles BKN and CLK to the area of triangle ABC are equal to 8/35 and
1/7 respectively, and ratio IK : AI is equal to 2/7. Find the ratio of the area of triangle
ANL to the area of triangle ABC.

Answer 1

Incentre $I$ divides the angle bisector in a defined ratio of the length of sides of the triangle

$$\dfrac{AI}{IK}=\dfrac{b+c}{a}$$ $$\dfrac{BI}{IL}=\dfrac{a+c}{b}$$

Let $\dfrac{ANL}{ABC} =X, AN=x, BN=y, AL=p, CL=q, CK=v, BK=u.$

An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

$$\dfrac{x}{y}=\dfrac{b}{a}, x+y=c$$ $$\dfrac{q}{p}=\dfrac{a}{c}, q+p=b$$ $$\dfrac{u}{v}=\dfrac{c}{b}, u+v=a$$

Then

$$x=\dfrac{bc}{a+b}, y=\dfrac{ac}{a+b}$$ $$u=\dfrac{ac}{b+c}, v=\dfrac{ab}{b+c}$$ $$p=\dfrac{bc}{a+c}, q=\dfrac{ab}{a+c}$$

The ratios of areas of triangles ANL, BKN and CLK to the area of triangle ABC are equal to X , 8/35 and 1/7 respectively. 

$$\dfrac{px}{bc}=X$$ $$\dfrac{uy}{ac}=\dfrac{8}{35}$$ $$\dfrac{vq}{ab}=\dfrac{1}{7}$$

Then

$$\dfrac{bc}{a+c}\cdot \dfrac{bc}{a+b}=Xbc$$ $$\dfrac{ac}{b+c}\cdot \dfrac{ac}{a+b}=\dfrac{8}{35}ac$$ $$\dfrac{ab}{b+c}\cdot \dfrac{ab}{b+c}=\dfrac{1}{7}ab$$

Let $\dfrac{b}{a}=z, \dfrac{c}{a}=w.$ We have

$$zw=X(1+w)(1+z)$$ $$w=\dfrac{8}{35}(z+w)(1+z)$$ $$z=\dfrac{1}{7}(z+w)(1+w)$$ $$\dfrac{8}{35}(z+w)(1+z)\cdot\dfrac{1}{7}(z+w)(1+w)=X(1+w)(1+z)$$ $$(z+w)^2=\dfrac{245X}{8}$$ $$z=\sqrt\dfrac{245X}{8}-w$$

Substitute

$$w=\dfrac{8}{35}(\sqrt\dfrac{245X}{8})(1+\sqrt\dfrac{245X}{8}+w)$$ $$w=\dfrac{8}{35}(\sqrt\dfrac{245X}{8}) +7X+ \dfrac{8}{35}(\sqrt\dfrac{245X}{8})w$$

$w= \dfrac{\dfrac{8}{35}(\sqrt\dfrac{245X}{8}) +7X}{1-\dfrac{8}{35}(\sqrt\dfrac{245X}{8}) }$

$z=\sqrt\dfrac{245X}{8}-\dfrac{\dfrac{8}{35}(\sqrt\dfrac{245X}{8}) +7X}{1-\dfrac{8}{35}(\sqrt\dfrac{245X}{8}) }$

$$\dfrac{AI}{IK}=\dfrac{b+c}{a}=z+w=\sqrt\dfrac{245X}{8}=7/2$$ $$\dfrac{245X}{8}=49/4 , X=0,4 \ Answer: \dfrac{ANL}{ABC}=0.4/1=2/5$$