# Triangle Explorations

 Please enable Java for an interactive construction (with Cinderella). Please enable Java for an interactive construction (with Cinderella).

### Things to try

• Construct the bisectors of the internal angles of triangle; then drag the triangle. Note that the bisectors appear to meet in a point. This point (the incentre) is the centre of the incircle of the triangle (the circle inside the triangle tangent to all three sides). Use Cinderella to construct this circle. There are three other circles tangent to all three sides; figure out how to construct them. (Hint: Angle bisectors still play a role.)
• Construct the altitudes of a triangle, and note that they appear to meet at a point (the orthocentre). Determine experimentally (by dragging) for which types of triangle the orthocentre is inside/outside the triangle. When the triangle is dragged, at what point does the orthocentre cross over the boundary between inside and outside?
• The circumcentre is the centre of the circumcircle, the circle through through the vertices of the triangle. It is located at the the intersection of the perpendicular bisectors of the sides - construct this point and then construct the circumcircle. Now answer the questions posed for the orthocentre: for which triangles is the circumcentre inside/outside the triangle, and where does it cross over?
• The medians of the triangle (joining each vertex to the midpoint of the opposite side) also appear to meet at a point ( the centroid - construct and drag to check). Construct points which divide the sides in fourths, etc. and experiment with which pairs of lines from two of these to the opposite vertex meet the median to the third side. This is a special case of Ceva's theorem: if the product of the ratios into which the points divide the three sides (taken in cyclic order) is 1 then the lines to these points from the opposite vertices meet in a point.
• Now construct the orthocentre, circumcentre and centroid on the same diagram. Make a conjecture about these points.
• Make arbitrary points on the sides of the triangle. Then construct three circles, each through a vertex and the two new points on the sides adjacent to it. Drag the triangle and your three points: what do you notice about the circles? (This is called Miquel's theorem, and is true even if the sides of the triangle are themselves parts of circles.)

Below is a listing of the objects in the sketch, their definitions, and their coordinates/equations. This list is updated dynamically as objects are moved or constructed.

 Please enable Java for an interactive construction (with Cinderella).