1. ## Triangles

Hello,

I need the solution of the following question:

Consider a triangle $\triangle ABC$ and let $X$ be a point on the side $AB$ such that $AX=\frac{1}{3}AB$ and let $Y$ be a point on the side $AC$ such that $CY=\frac{1}{3}AC$. Prove that the area of the triangle $\triangle AXC$ equals the area of the triangle $\triangle BYC$.

Best Regards.

2. Hello, raed!

$\text{Consider }\Delta ABC\text{; let }X\text{ be a point on side }AB\text{ such that: }AX\,=\,\frac{1}{3}AB$
$\text{and let }Y\text{ be a point on the side }AC\text{ such that }CY\,=\,\frac{1}{3}AC$.

$\text{Prove that the area of }\Delta AXC\text{ equals the area of }\Delta BYC.$

Side $\,AB$ is divided in the ratio $1:2$

Compare $\Delta AXC$ and $\Delta ABC.$

They have the same height $\,h.$

Code:
                    C
o
**|*
* * | *
*  *  |  *
*   *   |h  *
*    *    |    *
*     *     |     *
*      *      |      *
A o * * * o * * * * * * * o B
: - 1 - X - - - 2 - - - :

The base of $\Delta AXC$ is one-third the base of $\Delta ABC.$

Hence: . $(\text{area }\Delta AXC) \;=\;\frac{1}{3}(\text{area }\Delta ABC)$

In a similar fashion, we prove that: . $\text{(Area }\Delta BYC)} \;=\;\frac{1}{3}(\text{area }\Delta ABC})$

And we're done!

3. Originally Posted by Soroban
Hello, raed!

Side $\,AB$ is divided in the ratio $1:2$

Compare $\Delta AXC$ and $\Delta ABC.$

They have the same height $\,h.$

Code:
                    C
o
**|*
* * | *
*  *  |  *
*   *   |h  *
*    *    |    *
*     *     |     *
*      *      |      *
A o * * * o * * * * * * * o B
: - 1 - X - - - 2 - - - :

The base of $\Delta AXC$ is one-third the base of $\Delta ABC.$

Hence: . $(\text{area }\Delta AXC) \;=\;\frac{1}{3}(\text{area }\Delta ABC)$

In a similar fashion, we prove that: . $\text{(Area }\Delta BYC)} \;=\;\frac{1}{3}(\text{area }\Delta ABC})$

And we're done!

Thank you very much for your reply.

Best Regards,