4. Let $C \in (AB)$ and points $P, M$, $T$ be the midpoints of segments $(AB), (AC)$ and $(MP)$. If $2 \cdot AC - BC = 40 \text{ cm}$, then determine the length of segment $[TC]$.

Daniela and Nicolae Stănică, Brăila
## Solution:

Case I: $P \in(B C), T \in(M C)$

$$
\begin{aligned}
& A C=2 x, C B=2 y \Rightarrow 2 x-y=20 \\
& P C=y-x \text { and } M P=y \\
& M T=\frac{y}{2} \Rightarrow T C=x-\frac{y}{2}=\frac{2 x-y}{2}=10 \text{ cm}
\end{aligned}
$$

Cases II: $P \in(B C), T \in(C P)$ and III: $P \in(A C)$ are studied similarly and yield $T C=10 \text{ cm}$