High School: Geometry
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
At the heart of it, the study of mathematics is the study of relationships. Before jumping into Romeo and Juliet, though, let's be clear that the relationships of interest in mathematics are those between mathematical objects. Not between two star-cross'd lovers.
But in terms of depth, Shakespeare's got nothing compared to the beauty with which Euclid explored the deep and intrinsic connections between circles and triangles. Yeah, we're talking about a triangle's inscribed and circumscribed circles.
Students should know that an inscribed circle is the largest circle that can fit on the inside of a triangle, with the three sides of the triangle tangent to the circle. A circumscribed circle is one that contains the three vertices of the triangle. Students should know the difference between and be able to construct both of these circles.
Students should understand that while circles have one defined center, triangles try to outdo them by having four different centers: the incenter, the circumcenter, the centroid, and the orthocenter. Imitation is the sincerest form of flattery and all, but do you think maybe they're compensating for something?
Students should know the definitions of these centers and how each one differs from the others. But how do we get to the true center of the triangle? The world may never know.
We can draw any triangle by plotting three points on a circle. Students can take this one point further, plot four points on a circle, connect them, and make a cyclic quadrilateral. (They're called that because the vertices are all on the circle, in kind of a, well, cycle.)
Students be able to prove theorems about cyclic quadrilaterals, the most important of which is that opposite angles in a cyclic quadrilateral add to 180°. Or, if students are wearing their fancy pants today, they can say that opposite angles in a cyclic quadrilateral are supplementary. We would say that too, but our fancy pants are at the dry cleaners.
This only touches on the deep and fascinating love triangle between circles, triangles, and quadrilaterals. So maybe these shapes are less Romeo and Juliet and more A Midsummer Night's Dream. Well, the Lysander-Helena-Demetrius bit, anyway.