Problem 1)

https://www.geogebra.org/m/MAdsQxwT

In the figure, we have an arbitrary triangle ABC with altitudes AH_{a} , CH_{c} , BH_{b}.

Consider the following image above in GeoGebra. In order to prove Pythagoras theorem, I shall prove a stronger statement:

Namely, AB * BH_{c} + AC * CH_{C} = BC^{2}

Define angles a,b,c to represent angles A,B,C respectively. Now,in triangle BCH_{b} BH_{c} = BCcosb ,since angle H_{c} is a right angle. Recall lines CH_{c} ,BH_{b} and AH_{a} are altitudes.Similiarly, CH_{b} = BCcosc , BH_{a} = ABcosb and H_{a}C = ACcosc.

Now we have AB*BH_{c} + AC*CH_{b} = AB*BCcosb + AC*BCcosc

= BC*(AB.cosb + AC.cosc)

= BC*(BH_{a} + CH_{A})

= BC * BC

= BC^{2}

We have just proven the general statement.Now if Angle A is a right angle, the altitude BH_{b} will be the line AB and the altitude CH_{C} will be the line AC. Thus, AB = BH_{c} and AC = CH_{b}

This leads up to AB^{2} + AC^{2} = BC^{2}, which is the familiar Pythagoras theorem!