Cyclic constructions on triangles
Create in a snap cyclically defined objects in a triangle
A feature of OK Geometry allows you to quickly create cyclic constructions on triangles. The generated figures can be also easier to understand than the usual constructions.
A very simple example of a cyclic construction on a triangle is the construction of its centroid. Below is a commented construction of the centroid of a triangle that uses the cyclic construction feature.
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
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
It turns out that the points A', B', C', A'', B'', C'' lay on the same circle called the Dao 6-point circle.
Animation
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