Given:
![\[3x\text{ }+\text{ }2y\text{ }+\text{ }6\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-720e52ed7dd6bc49dab9c9629f25d973_l3.png "Rendered by QuickLaTeX.com")
![\[2x\text{ }-\text{ }5y\text{ }+\text{ }4\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aff186fbb14f8ff0cd2e2638a4f39091_l3.png "Rendered by QuickLaTeX.com")
And
![\[x\text{ }-\text{ }3y\text{ }-\text{ }6\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9c70ced6292de97d4266709c84e635d6_l3.png "Rendered by QuickLaTeX.com")
Let us assume, in triangle ABC, let equations (1), (2) and (3) represent the sides AB, BC and CA, respectively.
Solving equations (1) and (2), we have
![\[x\text{ }=\text{ }-2,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-162ea9d8960042cfcdce8f392e15753e_l3.png "Rendered by QuickLaTeX.com")
![\[~y\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-722e8eb47c61ee9889fd9281b7653635_l3.png "Rendered by QuickLaTeX.com")
Thus, AB and BC intersect at
![\[B\text{ }\left( -2,\text{ }0 \right).\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f16cd7786421673c25c2ce7b1b68e0f3_l3.png "Rendered by QuickLaTeX.com")
Now, solving (1) and (3), we have
![\[x\text{ }=\text{ }\text{ }6/11,\text{ }y\text{ }=\text{ }\text{ }24/11\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-924a6ce27e5dd7c9d7d2513606a628b5_l3.png "Rendered by QuickLaTeX.com")
Thus, AB and CA intersect at
![\[A\text{ }\left( -6/11,\text{ }-24/11 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-53361af6549515116c3f1c9160b6e8b7_l3.png "Rendered by QuickLaTeX.com")
Similarly, solving (2) and (3), we get
![\[x\text{ }=\text{ }-42,\text{ }y\text{ }=\text{ }-16\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc13c283f55aa85b96534aa88f1359e9_l3.png "Rendered by QuickLaTeX.com")
Thus, BC and CA intersect at
![\[C\text{ }\left( -42,\text{ }-16 \right).\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-63d0b45d58acdf3af6fbaa3097cb2721_l3.png "Rendered by QuickLaTeX.com")
Now,
let D, E and F be the midpoints the sides
BC, CA and AB, respectively.
Then, we have:
∴ The equations of the medians of a triangle are:
![\[41x\text{ }\text{ }112y\text{ }\text{ }70\text{ }=\text{ }0,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7eca86af3699e74a45add0d9b276655b_l3.png "Rendered by QuickLaTeX.com")
![\[16x\text{ }\text{ }59y\text{ }\text{ }120\text{ }=\text{ }0,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3c86b28edf987c1bc2680538ccf58475_l3.png "Rendered by QuickLaTeX.com")
And,
![\[25x\text{ }\text{ }53y\text{ }+\text{ }50\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fd313bb82858ffb1d827c753a148d096_l3.png "Rendered by QuickLaTeX.com")