Problem The bisector $AE$ of the apex angle $∠A$ of a triangle $ABC$ lies between the median line $AM$ and the perpendicular $AD$ drawn from this apex angle to the opposite side. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem In $△ABC$, suppose $AC>AB$, and if we draw the perpendicular $AD$ from $A$ to $BC$, we have $$∠DAC>∠DAB \qquad and \qquad DC>DB$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Since $AD⊥BC$,…

Problem The sum of the lengths of perpendiculars drawn from the three vertices of a triangle to their opposite sides is less than the sum of the lengths of the three sides of the triangle. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The angle made by perpendicular lines drawn from two vertices of an acute triangle to their opposite sides is equal to the supplementary angle of the remaining vertex. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…