circle. We call it the circle of Apollonius. This circle connects interior and exterior angle theorem, I and E divide AB internally and externally in the ratio k. Locus of Points in a Given Ratio to Two Points: Apollonius Circles Theorem. Apollonius Circle represents a circle with centre at a and radius r while the second THEOREM 1 Let C be the internal point of division on AB such that. PB.
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We illustrate how the reader can use the results of this encyclopedia and other computer-generated results. The computer-generated results, quoted below and not included in the first edition, will be included in the next editions.
The Apollonius circle of a triangle is the circle tangent internally to each of the three excircles. There are many ways the Apollonius apollojius to be constructed by using straightedge and compass. The famous Apollonius problem for three circles states: Given three apolllnius circles, to construct the circles tangent to each of them.
There are a few methods to solve the problem.
geometry – Apollonius circles theorem proof – Mathematics Stack Exchange
Dekov Software Geometric Constructions. To construct the Apollonius circle we can use one of these methods.
There are a few additional ways to construct the Apollonius circle. These additional methods are based on the fact that the given circles are not arbitrary, but they are the excircles of a given triangle. We shall see a few such methods below. Circel reader may consult Dekov Software Geometric Constructions for detailed description of constructions. Construct the center and the radius If we can construct the center and the radius of a circle, we can circcle the circle.
From page Theorems, Points, Center of the Apollonius Circlewe see that we can construct the center of the Apollonius circle as the intersection point of the Apollonius line and the Brocard axis known result, see Kimberling’s ETC .
Circles of Apollonius – Wikipedia
It known that the radius of the Apollonius circle is equal to M. Stevanovic  We can construct the radius. Hence, apollojius can construct the Apollonius circle. Note that in appllonius methods below we have not to construct the excircles. Construct the center and a point on the circle We can construct the center of the Apollonius circle see the previous section. Then we can construct one excircle, e. The intersection point of the line passing through the center of the Apolllonius circle and the A-Excircle lies on the Apollonius circle computer generated result; this result is obvious.
Construct three points of the circle If we can construct three points of a circle, then we can construct the circle as the circle passing through these three points. The three tangency points of the Apollonius circle and the excircles are the vertices of the Apollonius triangle. Hence, we can try to construct the Apollonius triangle, and then to construct its circumcircle, that is, the Circ,e circle.
There are many methods to construct a triangle. We can try to use the following method: If a triangle is perspective with two other triangles and if we can construct these triangles and the respective perspectors, then we can construct the triangle. The Apollonius triangle is perspective with 1 triangle ABC; the perspector is the Apollonius point this is the definition of the Apollonius point ; 2 Feuerbach triangle; the perspector is the Spieker center computer-generated result, not included in the first edition of this encyclopedia; is this result known?
From page Theorems, Points, Apollonius Pointwe can see a few ways to construct of the Apollonius point: F – Second Feuerbach point. P – Isotomic conjugate of F. A 1 B 1 C 1 – Apollonius triangle. S – Symmedian point. We can construct the Apollonius triangle by using any pair of triangles listed above.
The four triangles give us 6 ways to construct the Apollonius triangle. An extended computer research would give us probably a few additional triangles. Construct the Apollonius point X and the Spieker center S. Now we can construct the Apollonius circle as the circumcircle of the Apollonius triangle.
S – Spieker center. X – Apollonius point. A 1 B 1 C 1 – Feuerbach triangle. A 2 B 2 C 2 – Apollonius triangle. Apollonius circle as the inverse image of a circle A theorem from page Theorems, Circles, Apollonius Circle states that the Apollonius circle is the inverse of the Nine-point circle with respect to the radical circle apo,lonius the excircles.
The above result is known P. Yiu but the next result probably is not known. If we ask throrem computer to make a deeper investigation, we will receive the following result not included in the first edition of this encyclopedia; is this result known?
The Apollonius circle is congruent to the inverse circle of the Bevan circle with respect to the radical circle of the excircles of the anticomplementary triangle. Label by c the inverse circle of the Bevan circle with respect to the radical circle of the excircles of the anticomplementary triangle. Now we need the relationship between two points: We ask again the computer and receive a few relationships, e. Now we can construct the Apollonius circle as follows.
First, construct circle c. Then construct the center of the Apollonius circle as the midpoint of the incenter and the anticomplement of the center of circle c. Finally, construct the Apollonius circle. Ja, Jb, Jc – centers of the excircles.
The red triangle – Anticomplementary triangle. The three points on circle c are the inverse images of Ja, Jb, Jc with respect to circle cR. First we construct these three points, then we construct circle c as the circle passing through these points.
K – center of circle c. P – anticomplement of K. The use of the similitude centers of the Apollonius circle and another circle If we can construct a circle and the two similitude centers of the circle and the Apollonius circle, we can construct the Apollonius circle as follows.
First construct the center of the Apollonius circle as the harmonic conjugate of the center of the constructed circle with respect to the similitude centers.
Then construct the Apollonius circle. For example, we can use the circumcircle. The similitude centers could be constructed as follows: Construct the internal similitude center of the circumcircle and the Apollonius circle as the intersection point of the line passing through the circumcenter and the symmedian point the Brocard axisand the line passing through the orthocenter and the mittenpunkt.
Construct the external similitude center of the circumcircle and the Apollonius circle as the intersection point of the line passing through the incenter and the centroid theoerm Nagel lineand the line passing through the circumcenter and the symmedian point the Brocard axis.
Now construct the center of the Apollonius circle as the harmonic conjugate of the circumcenter with respect to the similitude centers, theorwm then construct the Apollonius circle. We can use a number of other circles in the place of the circumcircle. Concluding Remarks The methods above could be summarized to the following general method.
If we have to construct a geometric object, first we have to ask computer about the properties of this object. A computer program can answer the question.
Circles of Apollonius
Apolloniuz we need some additional information, we can ask again, and so on. Then we can use the properties to construct the object. All above constructions could be obtained by this way. Hence, we can replace the methods used by now inventiveness with a new method – questions and answers. In fact, the computer will solve the problem for us. Kimberling, Encyclopedia of Triangle Centers, available at http: Stevanovic, The Apollonius circle and related triangle centers, Forum Geometricorum, vol.
Yiu, Hyacintos messageJanuary 1,