Some Properties of the Harmonic Quadrilateral

: In this article, we review some properties of the harmonic quadrilateral related to triangle simedians and to Apollonius circles.


Definition 1. A convex circumscribable quadrilateral
having the property is called harmonic quadrilateral.

Definition 2.
A triangle simedian is the isogonal cevian of a triangle median. Proposition 2. In an harmonic quadrilateral, the diagonals are simedians of the triangles determined by two consecutive sides of a quadrilateral with its diagonal.
Proof. Let be an harmonic quadrilateral and (see Fig. 1). We prove that is simedian in the triangle . From the similarity of the triangles and , we find that: From the similarity of the triangles şi , we conclude that: From the relations (1) and (2) is an harmonic quadrilateral.

Proposition 4. In a triangle
, the points of the simedian of A are situated at proportional lengths to the sides and .
Proof. We have the simedian in the triangle (see Fig. 2). We denote by and the projections of on and respectively. We get: Moreover, from Proposition 1, we know that Substituting in the previous relation, we obtain that: On the other hand, From and , hence: If M is a point on the simedian and and are its projections on , and respectively, we have: Taking (4) into account, we obtain that: Remark 3. The converse of the property in the statement above is valid, meaning that, if is a point inside a triangle, its distances to two sides are proportional to the lengths of these sides. The point belongs to the simedian of the triangle having the vertex joint to the two sides. Proof. Let be an harmonic quadrilateral and any point within. We denote by the distances of to the , , , sides of lenghts (see Fig. 3). Let be the quadrilateral area. We have: This is true for and real numbers. Following Cauchy-Buniakowski-Schwarz Inequality, we get: and it is obvious that: We note that the minimum sum of squared distances is:

In
Cauchy-Buniakowski-Schwarz Inequality, the equality occurs if: Since is the only point with this property, it ensues that , so has the property of the minimum in the statement.
Definition 3. We call external simedian of triangle a cevian corresponding to the vertex , where is the harmonic conjugate of the point -simedian's foot from relative to points and . Fig. 4, the cevian is an internal simedian, and is an external simedian. We have:

Remark 4. In
In view of Proposition 1, we get that: Proposition 7. The tangents taken to the extremes of a diagonal of a circle circumscribed to the harmonic quadrilateral intersect on the other diagonal.
Proof. Let be the intersection of a tangent taken in to the circle circumscribed to the harmonic quadrilateral with (see Fig. 5). Since triangles PDC and PAD are alike, we conclude that: From relations (5), we find that: This relationship indicates that P is the harmonic conjugate of K with respect to A and C, so is an external simedian from D of the triangle . Similarly, if we denote by the intersection of the tangent taken in B to the circle circumscribed with , we get: From (6) and (7), as well as from the properties of the harmonic quadrilateral, we know that: which means that: hence . Similarly, it is shown that the tangents taken to A and C intersect at point Q located on the diagonal .

INTERNATIONAL FRONTIER SCIENCE LETTERS(IFSL)
ISSN: 2349-4484 Vol. is the internal simedian of the triangle and the property is proven. Remark 7. From this, in view of Fig. 5, it results that the circle of center Q passing through A and C is an Apollonius circle relative to the vertex A for the triangle . This circle (of center Q and radius QC) is also an Apollonius circle relative to the vertex C of the triangle . Similarly, the Apollonius circles corresponding to vertexes B and D and to the triangles ABC, and ADC respectively, coincide; we can formulate the following: Proposition 11. In an harmonic quadrilateral, the Apollonius circles -associated with the vertexes of a diagonal and to the triangles determined by those vertexes to the other diagonal -coincide. Radical axis of the Apollonius circles is the right determined by the center of the circle circumscribed to the harmonic quadrilateral and by the intersection of its diagonals. Proof. Referring to Fig. 5, we observe that the power of O towards the Apollonius circles relative to vertexes B and C of triangles and is: So O belongs to the radical axis of the circles. We also have , relatives indicating that the point K has equal powers towards the highlighted Apollonius circles.