Notes
three isoscelese triangles inside a triangle solution

Three Isosceles Triangles Inside a Triangle

Solution by Angles at a Point, Angles on a Straight Line, Angles in a Triangle, Angles in a Quadrilateral

With the points labelled as in the diagram, let $a$ be angle $E \hat{O} A$, $b$ angle $B \hat{O} D$, and $c$ angle $A \hat{O} B$. Then as angles at a point add up to $360^\circ$, $c = 360^\circ - 35^\circ - a - b$.

Triangle $E O A$ is isosceles so angle $A \hat{E} O$ is $\frac{180^\circ - a}{2} = 90^\circ - \frac{a}{2}$ as angles in a triangle add up to $180^\circ$. So since angles at a point on a straight line also add up to $180^\circ$, angle $O \hat{E} C$ is $90^\circ + \frac{a}{2}$. A similar argument shows that angle $C \hat{D} O$ is $90^\circ + \frac{b}{2}$.

Since the angles in a quadrilateral add up to $360^\circ$, considering quadrilateral $O D C E$ gives the equation:

and this simplifies to $a + b = 190^\circ$. Hence $c = 135^\circ$.