tkz-euclide

Points

Now that the fixed points are defined, we can get new points from their references, with the macros of the package or with macros you create yourself. The calculations are usually done by the package and remain hidden. You may need some mathematical constants; here is the list of the constants defined by the package.

7 Auxiliary tools¶

7.1 Constants¶

tkz-euclide knows the following constants:

7.2 New point by calculation¶

When a macro of tkz-euclide creates a new point, it is stored internally with the reference tkzPointResult. You can give it your own reference with the macro \tkzGetPoint; the new reference is placed between braces.

Sometimes you need to get two points. This is done with

If you only need the first or the second point, you can also use:

Sometimes the results consist of a point and a dimension. You get the point with \tkzGetPoint and the dimension with \tkzGetLength.

8 Special points¶

Here are some special points.

8.1 Midpoint of a segment \tkzDefMidPoint¶

This macro defines the midpoint of a segment.

8.1.1 Use of \tkzDefMidPoint¶

See also the use of \tkzDefPoint.

8.2 Golden ratio \tkzDefGoldenRatio¶

From Wikipedia: In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities \(a\) and \(b\) such that \(a > b > 0\), \(a+b\) is to \(a\) as \(a\) is to \(b\).

\( \frac{a+b}{a} = \frac{a}{b} = \phi = \frac{1 + \sqrt{5}}{2}\)

One of the two solutions of the equation \(x^2 - x - 1 = 0\) is the golden ratio \(\phi\), \(\phi = \frac{1 + \sqrt{5}}{2}\).

8.2.1 Use the golden ratio to divide a line segment¶

8.2.2 Golden arbelos¶

You can also use the following macro.

8.3 Barycentric coordinates with \tkzDefBarycentricPoint¶

\(n\) points \(pt_1\), \(pt_2\), …, \(pt_n\) define \(n\) vectors \(\overrightarrow{v_1}\), \(\overrightarrow{v_2}\), …, \(\overrightarrow{v_n}\) with the origin as common starting point. \(\alpha_1\), \(\alpha_2\), …, \(\alpha_n\) are \(n\) numbers; the vector

defines a unique point.

8.3.1 With two points¶

In the following example, we get the barycenter of the points \(A\) and \(B\) with the coefficients \(1\) and \(2\), in other words:

\[ \overrightarrow{AI}= \frac{2}{3}\overrightarrow{AB} \]

8.3.2 With three points¶

This time \(M\) is simply the centroid of the triangle.

For simplicity and consistency, there is also \tkzCentroid.

8.4 Internal and external similitude centers: \tkzDefSimilitudeCenter¶

The centers of the two homotheties that map one circle onto another are called the external and internal centers of similitude. You can use \tkzDefIntSimilitudeCenter and \tkzDefExtSimilitudeCenter, but the following macro is better.

8.4.1 Internal and external with node¶

8.4.2 D’Alembert Theorem¶

You can also use \tkzDefBarycentricPoint to find a center of homothety:

\tkzDefBarycentricPoint(O=\r,A=\R) \tkzGetPoint{I}
\tkzDefBarycentricPoint(O={-\r},A=\R) \tkzGetPoint{J}

8.4.3 Example with node¶

8.5 Harmonic division with \tkzDefHarmonic¶

8.5.1 Options ext and int¶

8.5.2 Bisector and harmonic division¶

8.5.3 Option both¶

both is the default option

8.6 Equidistant points with \tkzDefEquiPoints¶

8.6.1 Using \tkzDefEquiPoints with options¶

8.7 Midpoint of an arc¶

9 Point on line or circle¶

9.1 Point on a line with \tkzDefPointOnLine¶

9.1.1 Use of option pos¶

9.2 Point on a circle with \tkzDefPointOnCircle¶

The order of the arguments has changed: now it is center, angle and point or radius. Two options work with radians: through in rad and R in rad.

9.2.1 Altshiller’s Theorem¶

The two lines joining the points of intersection of two orthogonal circles to a point of one of the circles meet the other circle in two diametrically opposite points (Altshiller-Court, p. 176).

9.2.2 Use of \tkzDefPointOnCircle¶

10 Special points relating to a triangle¶

10.1 Triangle center: \tkzDefTriangleCenter¶

10.1.1 Option ortho or orthic¶

The intersection \(H\) of the three altitudes of a triangle is called the orthocenter.

10.1.2 Option centroid¶

10.1.3 Option circum¶

10.1.4 Option in¶

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle’s incenter. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex \(A\), for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex \(A\), or the excenter of \(A\). Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.
(Article on Wikipedia)

We get the center of the incircle of the triangle. The result is of course in tkzPointResult; we retrieve it with \tkzGetPoint.

10.1.5 Option ex¶

An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle’s sides.
(Article on Wikipedia)

We get the center of an excircle of the triangle. The result is of course in tkzPointResult; we retrieve it with \tkzGetPoint.

10.1.6 Option euler¶

This macro gives the center of the nine-point circle, also called Euler’s circle or Feuerbach’s circle. The nine-point circle, also called Euler’s circle or the Feuerbach circle, is the circle that passes through the perpendicular feet \(H_A\), \(H_B\), and \(H_C\) dropped from the vertices of any reference triangle \(ABC\) on the sides opposite them. Euler showed in 1765 that it also passes through the midpoints \(M_A\), \(M_B\), \(M_C\) of the sides of \(ABC\). By Feuerbach’s theorem, the nine-point circle also passes through the midpoints \(E_A\), \(E_B\), and \(E_C\) of the segments that join the vertices and the orthocenter \(H\). These points are commonly referred to as the Euler points.
(https://mathworld.wolfram.com/Nine-PointCircle.html)

10.1.7 Option symmedian¶

The point of concurrence \(K\) of the symmedians, sometimes also called the Lemoine point (in England and France) or the Grebe point (in Germany).
Weisstein, Eric W. "Symmedian Point." From MathWorld–A Wolfram Web Resource.

10.1.8 Option spieker¶

The Spieker center is the center \(Sp\) of the Spieker circle, i.e., the incenter of the medial triangle of a reference triangle.
Weisstein, Eric W. "Spieker Center." From MathWorld–A Wolfram Web Resource.

10.1.9 Option gergonne¶

The Gergonne Point is the point of concurrency which results from connecting the vertices of a triangle to the opposite points of tangency of the triangle’s incircle. (Joseph Gergonne, French mathematician.)

10.1.10 Option nagel¶

Let \(Ta\) be the point at which the excircle with center \(Ja\) meets the side \(BC\) of a triangle \(ABC\), and define \(Tb\) and \(Tc\) similarly. Then the lines \(ATa\), \(BTb\), and \(CTc\) concur in the Nagel point \(Na\).
Weisstein, Eric W. "Nagel point." From MathWorld–A Wolfram Web Resource.

10.1.11 Option mittenpunkt¶

The mittenpunkt (also called the middlespoint) of a triangle \(ABC\) is the symmedian point of the excentral triangle, i.e., the point of concurrence \(M\) of the lines from the excenters through the midpoints of the corresponding sides.
Weisstein, Eric W. "Mittenpunkt." From MathWorld–A Wolfram Web Resource.

10.1.12 Relation between gergonne, centroid and mittenpunkt¶

The Gergonne point \(Ge\), the centroid \(G\) and the mittenpunkt \(M\) are collinear, with \(GeG/GM=2\).