THE GERGONNE POINT GENERALIZED THROUGH CONVEX COORDINATES

The Gergonne point of a triangle is the point at which the three cevians to the points of tangency between the incircle and the sides of the triangle are concurrent. In this paper, we follow Koneĉný [7] in generalizing the idea of the Gergonne point and find the convex coordinates of the generalized Gergonne point. We relate these convex coordinates to the convex coordinates of several other special points of the triangle. We also give an example of relevant computations.


Introduction.
When cevians of particular significance in the general triangle (medians, angle bisectors, etc.) are concurrent, their common point is often called a special point of the triangle.Such points have always held interest for geometers.In the past, we have discovered the convex coordinates [7,6] of several special points at which cevians from the three vertices are concurrent [3,2].We choose the terminology "convex coordinates" rather than the more widely used "barycentric" or "trilinear coordinates" because of their relevance to convex sets [9].Now, let the circle C(I) be the incircle of V 1 V 2 V 3 as shown below.The cevians from the vertices to the points of tangency on the opposite sides of the triangle are concurrent at the point G which is known as the Gergonne point of the triangle [5,8].
The convex coordinates of a point P in the plane of V 1 V 2 V 3 and relative to this triangle may be taken as weights which if placed at vertices V 1 , V 2 , V 3 , cause P to become the balance point for the plane.The plane is taken to be horizontal and otherwise weightless.Also, we require that the sum of the weights have unit value.If P belongs to the closed triangular region, then all the three weights are nonnegative.
We denote the weight placed at vertex V i by α i .Then point P has convex coordinates (α 1 ,α 2 ,α 3 ) with respect to V 1 , V 2 , V 3 , in that order, and α 1 + α 2 + α 3 = 1.For example, the convex coordinates of the vertices V 1 , V 2 , V 3 , are (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively, and the convex coordinates of the centroid of Let us return to the Gergonne point G and consider V 1 V 2 V 3 with its incircle C(I) as redrawn in Figure 2. The point at which C(I) touches the side opposite to V i is denoted by A i .The points of tangency divide the sides into segments of lengths x 1 ,x 2 ,x 3 as shown in the figure.
It is not a difficult task to show that the convex coordinates of G have the values where The lengths of the sides and measures of the angles of V 1 V 2 V 3 are more immediately accessible numbers than are x 1 ,x 2 , and x 3 .So, let i denote the length of the side opposite to vertex V i and θ i denote the measure of the angle at V i .Figure 3 depicts the triangle again and establishes the notation for the work to follow.
3 with sides and angles labeled.
The lengths of the sides and the values x 1 ,x 2 , and x 3 are related by the equations (2) Thus, Substitution of these results into the expressions (1) for the convex coordinates of the Gergonne point would not improve their already pleasing appearance.
We have found that convex coordinates provide a straightforward method for investigating special points of triangles.Thus, we read with interest a problem proposed by V. Koneĉný [7] which concerns a generalization of the Gergonne point.In our paper, we find the convex coordinates of Koneĉný's generalized Gergonne point and, in the process, provide an independent proof that the relevant cevians are concurrent.

A generalization of the Gergonne point.
Let I be the incenter of V 1 V 2 V 3 and let D(I) be a circle concentric with incircle C(I) as shown in Figure 4. Suppose that lines are drawn through I perpendicular to the sides of the triangle.These lines intersect the sides of V 1 V 2 V 3 at A 1 ,A 2 ,A 3 , the points of tangency between the triangle and the incircle and they intersect circle D(I) at points B 1 ,B 2 , and B 3 .

and circle D(I).
Koneĉný's problem is to show that the cevians The point H at which the cevians are concurrent is a generalized Gergonne point.We compute the convex coordinates for H and the computational path to our result makes it obvious that the cevians are concurrent.
We begin by noting that there exists W 1 W 2 W 3 for which D(I) is the incircle and for which H is the Gergonne point.This triangle is similar to V 1 V 2 V 3 ; its sides are parallel to the corresponding sides of V 1 V 2 V 3 ; and the two triangles are "concentric".If the radius of C(I) is r > 0, then the radius of D(I) is r + t, where −r < t.The corresponding sides of the two triangles are a perpendicular distance of |t| units apart and the similarity ratio for lengths in the two triangles is (length in The geometry of the two triangles and their incircles is shown in Figure 5.
Points B 1 ,B 2 ,B 3 divide the sides of W 1 W 2 W 3 into segments of lengths y 1 ,y 2 ,y 3 just as A 1 ,A 2 ,A 3 divide the sides of V 1 V 2 V 3 into segments of lengths x 1 , x 2 , x 3 .An additional crucial observation is that W i , V i , I are collinear for i = 1, 2, 3 since rays → W i I and → V i I bisect the congruent vertex angles at W i and V i .
Let R i be the point at which These segments serve as lever arms when weights taken to define balance lines.We show the calculations for the length of V 1 R 3 in some detail and then simply state the other five lengths.
We extend V 1 V 3 to intersect W 1 W 2 at point P and we draw angle bisector W 1 I through V 1 as shown in Figure 6.
Since triangles W 1 W 2 W 3 and V 1 V 2 V 3 are similar with similarity ratio (r + t) : r , it should be clear that where the meaning of x i for i = 1, 2, 3 is given in Figure 2. The length of P B 3 is given by and the length of Before proceeding, let us simplify our notation by letting Then P B 3 = y 1 − tm 1 and P V 1 = tm 1 .
Triangles P V 3 B 3 and V 1 V 3 R 3 are also similar.Therefore, where Appropriate substitutions yield so that all values except t depend only upon the geometry of Letting m i = cot(θ i /2)−cot θ i for i = 1, 2, 3, we give the results of the other computations for lever arms At first glance, these expressions seem quite complicated but an examination of the subscripts should reveal their symmetry.The appearance of such symmetry gives confidence in the computations thus far.However, for the next calculations, a few changes in form are helpful.We make the substitutions suggested by equations ( 1) and (9).The result, denoted by (9), follows from the triangles shown in Figure 7.
We have extended Equations ( 7) and ( 8) become It becomes clear at this stage that V 1 R 1 , V 2 R 2 , and V 3 R 3 must be concurrent.That conclusion follows from Ceva's theorem since To find the convex coordinates of H, the generalized Gergonne point at which the cevians are concurrent, we place weights α 1 ,α 2 ,α 3 at V 1 ,V 2 ,V 3 respectively and require that the cevians V 3 R 3 and V 1 R 1 define balance lines.This means that and The weights are denoted by α i since we will not normalize coordinates until we have convinced ourselves that we have a triple of weights with point H as balance point.Then we write α i = α i /(α 1 + α 2 + α 3 ).
A bit of algebra suggests that The reader may satisfy himself that α 1 ,α 2 ,α 3 do indeed satisfy equations ( 12) and (13).The two cevians define balance lines and their point of intersection must be the balance point.The third cevian defines a balance line if and only if Substitution of the values for α 1 ,α 2 ,α 3 into equation ( 15) yields This last equation is valid by Ceva's theorem.Thus, the convex coordinates of the generalized Gergonne point H are given by where the definitions of α 1 ,α 2 ,α 3 are given by equation (14).

Three checks and an example
x 3 and 1 = 2 = 3 which implies that α 1 = α 2 = α 3 = 1/3 as desired since the symmetry of the triangle forces H to coincide with the centroid.Check 2. If t = 0, H becomes the Gergonne point.Letting t = 0 means that k = 0. Then equations ( 7), (8), and (14) imply that α 1 ,α 2 ,α 3 have the values given by equations (1).Check 3. If t → −r .Then H approaches the incenter I of ∆V 1 V 2 V 3 and the convex coordinates (α 1 ,α 2 ,α 3 ) approach the convex coordinates of the incenter,  ( These Cartesian coordinates can also be found by analytic geometry as a further check upon the accuracy of our computations.