Squares, Triangles, and Van-Aubel’s Theorem

Problem In the figure, H is the midpoint of side BC of ΔABC. The points I and J are the intersection of the diagonals of square ABDE and square ACGF respectively, that is they are the centers of the square. Prove that IH and HJ are congruent and that angle IHJ is a right angle.

What is the point of tangency – no calculus, please

Problem Two lines intersect the graph . The secant line intersects the graph at point A and B and the tangent line which is parallel to the secant, touches the graph at point E. You can read the coordinates of A and B from the graph but not E. What are the coordinates of E?

Minimum time to cross the river – a puzzle

Puzzle Jose, Carlos, Linda, and Anna need to cross a bridge. It was dark and they only have one torch which they need to use to cross the bridge. Each of them have crossed the bridge before alone. Jose can run and cross it for one minute. Carlos can also do it in two minutes.