Isogonic or
Jacobi's Theorem: Isogonals and Concurrent Point.
Level: High
School, SAT Prep, College
Through vertices of a given triangle
ABC, we draw two lines isogonal conjugates with respect to the
corresponded angle. Prove that the lines connecting each vertex
of the given triangle with the intersection point of the
isogonal lines through the other vertices, AA', BB', and CC' are
concurrent at one point J called the Jacobi Point.
You can interact with
the right figure by dragging the red points and lines.
The isogonal conjugate line to BC' is the line BA' obtained
by reflecting the line BC' on the angle bisector of angle B;
in the figure above: angles ABC' = CBA'. Similarly AC' and
AB' are isogonal conjugates (angles BAC' = CAB'), and CA'
and CB' are isogonal conjugates (angles ACB' = BCA').
Dynamic Geometry: You can alter the figure above
dynamically in order to test and prove (or disproved)
conjectures and gain mathematical insight that is less
readily available with static drawings by hand.
This page uses the
TracenPoche
dynamic geometry software and requires
Adobe Flash player 7 or higher.
TracenPoche is a project of Sesamath, an association of French
teachers of mathematics.
Instruction to explore the
illustration above:
Animation. Click the red
button
to start/stop animation
Manipulate. Drag points A
and C, and line AC to change the figure.
Step-by-Step construction.
Press P and click the left mouse
button
on any free area to show the
step-by-step bar and click 'Next
Step' button ()
to start the construction step-by-step:
Hide the step-by-step bar by
using again the combination P +
click left mouse.
to start/stop animation
)
to start the construction step-by-step:
