Cyclic constructions on triangles

Create in a snap cyclically defined objects in a triangle

We start the construction with the Special|A triangle command, which also sets the initial cyclic structure on the triangle ABC. We also turn ON the Cycl indicator with the command Special|Cyclic construction or by clicking on it.

Creating point A' as the midpoint of the segment BC also creates its cyclic counterparts, i.e. points B' and C'. Note that they appear bleached. The created points A', B', C' become part of the cyclic structure of the triangle.
 

When we create the line segment AA', we also create its cyclic counterparts, i.e., the line segments BB' and CC'. Note that they appear bleached.

We create the point G (the centroid G) as the intersection of the lines BB' and CC'. OK Geometry recognises that the two counterparts of G coincide with G, and they are thus ignored.

• The cyclic construction indicator Cycl (at the top of the display) can be switched ON or OFF with a click on it. 
• When the cyclic construction indicator Cycl is ON, the creation of an object generally creates also the two cyclic counterparts of that object. If a created object coincides with its cyclic counterparts, OK Geometry either ignores the cyclic counterparts or warns you of the coincidence by changing the shape of the cursor (so thatyou can eventually repeat the operation with Cycl OFF).
• Certain types of objects cannot be created using cyclic construction. You can also add a triplet of objects to the  cyclic structure.
• The cyclic counterparts are bleached by default. All or some of them can be un-bleached with appropriate commands.

Example 1

Here is an example of a cyclic construction:

Given is a triangle ABC. Let A' be the inverse of A in the incircle of the triangle ABC. Define B' and C' cyclically. Let A'' be the intersection, other that A, of the circle through A, B, C' and the circle through A, C, B'. Define B'' and C'' cyclically.

OK Geometry observes that the lines AA'', BB'', CC'' concur in a point P. A triangle analysis of the point shows that P is the ETC centre X10481 (perspector of ABC and cross-triangle of ABC and inverse-in-incircle triangle). In other words, the intersection of lines BC' and B'C is on the line AA'' and the same is true for the cyclic counterparts of this property.

Animation
 

Example 2 - The Dao 6-point cycle

Let G be the centroid of the triangle ABC. Let A' be the centre of the circle passing through B and tangent to the line AG at G. Define B' and C' cyclically. Let A'' be the centre of the circle that passes through C and is tangent to the line AG at G. Define B'' and C'' cyclically.

It turns out that the points A', B', C', A'', B'', C'' lay on the same circle called the Dao 6-point circle.

Animation

Example 3 - Detecting cyclic perspectivities

Let N be the nine-point centre of the triangle ABC. Let A' be the centre of the circle passing through points B, C and N. Define B' and C' cyclically. 

Before the point A' was constructed, the flag Cycl and the detection of cyclic perspectivities was activated. OK Geometry observed that the triangles ABC and A'B'C' are orthologically and cyclologically perspective.  Thus
  - the perpendicular through A' to BC,
- the perpendicular through B' to CA,
- the perpendicular through C' to AB
       meet at the point orth_ (X3 of ABC);

- the perpendicular through A to B'C',
- the perpendicular through B to C'A',
- the perpendicular through C to A'B'
       meet at the point orth (X5 of ABC);

- the circle through A', B, C,
- the circle through B', C, A,
- the circle through C', A, B
       meet at the point cycl_ (X24772 of ABC);

- the circle through A, B', C',
- the circle through B, C', A',
- the circle through C, A', B'
       meet at the point cycl_ (X1291 of ABC).

Animation