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- In an isosceles triangle $ABC$, where $AB = BC$, there exists a point $P$ on the segment $AC$ such that $AP = 6, P C = 4$, and $BP = 2$. Determine the perimeter of triangle $ABC$.
- A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle{MAC} = \angle{MCD} = x$. Determine $\angle{ABM}$.
- In the diagram $\triangle{MNP}$ with $R$ on $MP$ and $T$ on $MN$ is given such that $RT \parallel PN$. S is a point on $PN$ such that $TS \parallel MP$.
- O is the centre of the circle PTR. N is a point on chord RP such that ON $\perp$ PR. RS and PS are tangents to the circle at R and P respectively. RS=15 units; TS=9 units; $R\hat{P}S=42,83^{\circ}$.
- Prove that if a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.

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