# Nested Isosceles Triangles +-- {.image} [[NestedIsoscelesTriangles.png:pic]] > Both the outer black triangle and inner pink triangle are isosceles, and the three coloured areas are equal. What’s the angle? =-- ## Solution by [[Triangle area]], [[Angle in a Semi-Circle]], and [[Angles on a Straight Line]] +-- {.image} [[NestedIsoscelesTrianglesLabelled.png:pic]] =-- With the points labelled as in the above diagram, triangles $A E D$ and $D E B$ have the same area and the same height above the line $A B$, so the lengths of their bases along that line must be the same. That is, $A D$ and $D B$ have the same length. So a circle centred on $D$ that passes through $B$ will also pass through $E$ and $A$. The line segment $A B$ is then a diameter of that circle and $E$ a point on its circumference, so since the [[angle in a semi-circle]] is $90^\circ$, angle $A \hat{E} B$ is $90^\circ$. Then as [[angles at a point on a straight line]] add up to $180^\circ$, angle $B \hat{E} C$ is also $90^\circ$.