In right triangle $\mathrm{A}\mathrm{B}\mathrm{C},$ right angled at $\mathrm{C}, \mathrm{M}$ is the midpoint of hypotenuse $\mathrm{A}\mathrm{B}. \mathrm{C}$ is joined to $\mathrm{M}$ and produced to a point $\mathrm{D}$ such that $\mathrm{D}\mathrm{M}=\mathrm{C}\mathrm{M}.$ Point $\mathrm{D}$ is joined to point $\mathrm{B}$ as shown in figure. Show that $\mathrm{\angle }\mathrm{D}\mathrm{B}\mathrm{C}$ is a right angle.

 

In $\mathrm{\Delta A}\mathrm{B}\mathrm{C}, \mathrm{A}\mathrm{D}$ is the perpendicular bisector of $\mathrm{B}\mathrm{C}$ as shown in figure. Show that $\mathrm{\Delta } \mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle in which $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$

$\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle in which altitudes $\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ are drawn to equal sides $\mathrm{A}\mathrm{C}$ and $\mathrm{A}\mathrm{B}$ respectively as shown in figure. Show that these altitudes are equal.