Br0card P0oints

The Brocard Point (which has a twin brother simply named The Second Brocard Point) is a simple but complex idea in geometry that French Mathematician, Henri Brocard, discovered. In the simplest of words, it is a point (Q) in the triangle that, if an angle was made by drawing a line from the point Q, to a vertex and then another one, it would form a Brocard Angle. A Brocard Angle just means that if one were to repeat the same process three times in the same triangle connecting the points in the same respective manner, all three angles would be congruent. The Second Brocard Point is the point that would be found if one were to find the First Brocard Point using a backwards process. (In other words, you would be finding the three angles that are not identified on this geogebra file). This point allows us to view a relationship between the triangle and three circles (each on tangent to each of the three sides, and passes through two vertices), the division of a triangle into smaller triangles with a corresponding angle, and the sides of triangles (their perpendicular bisectors and lines that run through vertices). The FBP should not end up at vertex B of the triangle or should not be on the outside of the triangle. If you have configured the triangle in a manner that causes it to be, please reset the shape. (In other words, the triangle should always be read, Triangle CAB, ABC, or BCA clockwise.

1) Can you get the Brocard Point in the center of the triangle? Explain why or why not. 2) When you click the empty, white circle on lines d, g, and i (turning the circles blue0, the point that is the intersection of these lines (the medians of the triangle) is called the centroid. - When is the centroid and the FBP the same? - When are they furthest apart? 3) Repeat the same process for lines l, m, and n. The point of this intersection is called the incenter. - When is the incenter, centroid, and FBP the same? - When are they the most unrelated (furthest away or most spreaded out) in terms of placement in the triangle. 4) Click on the lines e, h, and j under the "Line" category in the Algebra box to display them. These lines are the perpendicular to a side respective side of the triangle, and pass through the center of one of the three circles. These lines form a triangle that encompasses the triangle with the FBP. - The four triangles that make up the entire construction appear to have the same area when...? 5) Show all of the Brocard Angles. - Make an observation about the three triangles. How are they different? Where are they the same?