Triangle, vectors, hard to explain

Question

P is the middle of a median line from vertex A, of ABC triangle. If Q is the point of intersection of lines AC and BP. Find relations of $|\vec{AQ}|$/$|\vec{QC}|$ and $|\vec{BP}|$/$|\vec{PQ}|$

Any suggestions for the title, welcomed.

Answer 1

We have $M=\tfrac12(B+C)$, $P=\tfrac12(A+M)$

Expressions for the points on the lines $AC$ and $BP$ in parametric form are:

\begin{align} p_{AC}(s) &= (1-s)\cdot A+s\cdot C, \\ p_{BP}(t) &= (1-t)\cdot B+t\cdot P. \end{align}

Since $Q=AC \cap BP$, \begin{align} Q&= p_{AC}(s) = p_{BP}(t), \\ (1-s)\cdot A+s\cdot C &= (1-t)\cdot B+t\cdot P. \end{align} The last one gives a linear system of two equations (for $x$ and $y$ coordinates) in two unknowns $s$ and $t$, with solution $s=\tfrac13$, $t=\tfrac43$ and then the answer is

\begin{align} \frac{|AQ|}{|QC|}&=\tfrac12 \\ \frac{|BP|}{|PQ|}&=3. \end{align}