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A New Theorem on Orthogonal Quadrilaterals

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A New Theorem on Orthogonal Quadrilaterals International Journal of Mathematics Research. ISSN 0976-5840 Volume 11, Number 1 (2019), pp. 5-14 © International Research Publication House http://www.irphouse.com A New Theorem on Orthogonal Quadrilaterals Brian Ekanyu Abstract In this paper, I prove an existence of a new theorem on orthogonal quadrilateralss. The theorem is stated as: The sum of the product of side, a and its altimedian distance, 휶 and the product of side, d and its altimedian distance, 휹 is equal to the sum of the product of side, b and its altimedian distance, 휷 and the product of side, c and its altimedian distance, 휸 i.e.: 휶풂 + 휹풅 = 휷풃 + 휸풄 INTRODUCTION Definitions Orthogonal quadrilateral: Is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other. B A E C D 6 Brian Ekanyu Altitude of a triangle: Is a line segment through a vertex and perpendicular to a line containing the base (the side opposite the vertex). This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called “the altitude,” is the distance between the vertex and the foot of the altitude. Q P M R Q- Is the vertex M- Is the foot of the altitude 푸푴̅̅̅̅̅ – Is the altitude of the triangle PQR A New Theorem on Orthogonal Quadrilaterals 7 Median of a triangle: Is a line segment joining a vertex to the mid-point of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex and they all intersect each other at the triangle’s centroid. V Y Z T U W X T-Is the centroid Altimedian distance: Is the distance between the foot of the altitude and the mid- point of the base line. R S 푹푺̅̅̅̅ – Is the altimedian distance 8 Brian Ekanyu Theorem: Consider an orthogonal quadrilateral ABCD as shown above such that: 푨푩̅̅̅̅=a, 풊풋̅ =휶 푩푪̅̅̅̅ = 풅, 푲푳̅̅̅̅ = 휹 푨푫̅̅̅̅ = 풄, 푮푯̅̅̅̅̅ = 휸 푪푫̅̅̅̅ = 풃, 푴푵̅̅̅̅̅ = 휷 Then 휶풂 + 휹풅 = 휷풃 + 휸풄 where 휶, 휷, 휸 풂풏풅 휹 are altimedian distances. A New Theorem on Orthogonal Quadrilaterals 9 Proof From triangle ABD 10 Brian Ekanyu 푨푩̅̅̅̅=a, 풊풋̅ = 휶 푨푫̅̅̅̅=c, 푮푯̅̅̅̅̅ = 휸 푩푫̅̅̅̅̅=f, 푬푭̅̅̅̅ = ∅ Considering triangle ABG 풄 푨푮̅̅̅̅ = ( − 휸) , 푩푮̅̅̅̅ = 풉 , 푨푩̅̅̅̅ = 풂 ퟐ ퟏ 푨푮̅̅̅̅ퟐ + 푩푮̅̅̅̅ퟐ = 푨푩̅̅̅̅ퟐ 풄 ퟐ ( − 휸) + 풉 ퟐ = 풂ퟐ ퟐ ퟏ 풄ퟐ 풉 ퟐ = 풂ퟐ − + 휸풄 − 휸ퟐ [ퟏ] ퟏ ퟒ Considering triangle BGD 풄 푩푮̅̅̅̅ = 풉 , 푮푫̅̅̅̅ = ( + 휸), 푩푫̅̅̅̅̅ = 풇 ퟏ ퟐ 푩푮̅̅̅̅ퟐ + 푮푫̅̅̅̅ퟐ = 푩푫̅̅̅̅̅ퟐ 풄 ퟐ 풉 ퟐ + ( + 휸) = 풇ퟐ ퟏ ퟐ 풄ퟐ 풉 ퟐ = 풇ퟐ − − 휸풄 − 휸ퟐ [ퟐ] ퟏ ퟒ Equating [ퟏ] and [ퟐ] 풄ퟐ 풄ퟐ 풂ퟐ − + 휸풄 − 휸ퟐ = 풇ퟐ − − 휸풄 − 휸ퟐ ퟒ ퟒ ퟐ휸풄 = 풇ퟐ − 풂ퟐ [ퟑ] Considering triangle ADI 풂 푨푰̅̅̅̅ = ( − 휶) , 푫푰̅̅̅̅ = 풉 , 푨푫̅̅̅̅ = 풄 ퟐ ퟐ 푨푰̅̅̅̅ퟐ + 푫푰̅̅̅̅ퟐ = 푨푫̅̅̅̅ퟐ 풂 ퟐ ( − 휶) + 풉 ퟐ = 풄ퟐ ퟐ ퟐ 풂ퟐ 풉 ퟐ = 풄ퟐ − + 휶풂 − 휶ퟐ [ퟒ] ퟐ ퟒ Considering triangle BDI 푩푰̅̅̅̅ퟐ + 푫푰̅̅̅̅ퟐ = 푩푫̅̅̅̅̅ퟐ 풂 푩푰̅̅̅̅ = ( + 휶) , 푫푰̅̅̅̅ = 풉 , 푩푫̅̅̅̅̅ = 풇 ퟐ ퟐ 풂 ퟐ ( + 휶) + 풉 ퟐ = 풇ퟐ ퟐ ퟐ A New Theorem on Orthogonal Quadrilaterals 11 풂ퟐ 풉 ퟐ = 풇ퟐ − − 휶풂 − 휶ퟐ [ퟓ] ퟐ ퟒ Equating [ퟒ] and [ퟓ] 풂ퟐ 풂ퟐ 풄ퟐ − + 휶풂 − 휶ퟐ = 풇ퟐ − − 휶풂 − 휶ퟐ ퟒ ퟒ ퟐ휶풂 = 풇ퟐ − 풄ퟐ [ퟔ] Considering triangle ADE 풇 푨푬̅̅̅̅ = 풉 , 푫푬̅̅̅̅ = ( + ∅), 푨푫̅̅̅̅ = 풄 ퟑ ퟐ 푨푬̅̅̅̅ퟐ + 푫푬̅̅̅̅ퟐ = 푨푫̅̅̅̅ퟐ 풇 ퟐ 풉 ퟐ + ( + ∅) = 풄ퟐ ퟑ ퟐ 풇ퟐ 풉 ퟐ = 풄ퟐ − − ∅풇 − ∅ퟐ [ퟕ] ퟑ ퟒ Considering triangle ABE 풇 푨푬̅̅̅̅ = 풉 , 푩푬̅̅̅̅ = ( − ∅), 푨푩̅̅̅̅ = 풂 ퟑ ퟐ 푨푬̅̅̅̅ퟐ + 푩푬̅̅̅̅ퟐ = 푨푩̅̅̅̅ퟐ 풇 ퟐ 풉 ퟐ + ( − ∅) = 풂ퟐ ퟑ ퟐ 풇ퟐ 풉 ퟐ = 풂ퟐ − + ∅풇 − ∅ퟐ [ퟖ] ퟑ ퟒ Equating [ퟕ] and [ퟖ] 풇ퟐ 풇ퟐ 풂ퟐ − + ∅풇 − ∅ퟐ = 풄ퟐ − − ∅풇 − ∅ퟐ ퟒ ퟒ ퟐ∅풇 = 풄ퟐ − 풂ퟐ [ퟗ] Adding [ퟑ], [ퟔ] and [ퟗ] ퟐ휸풄 + ퟐ휶풂 + ퟐ∅풇 = 풇ퟐ − 풂ퟐ + 풇ퟐ − 풄ퟐ + 풄ퟐ − 풂ퟐ ퟐ휶풂 + ퟐ휸풄 + ퟐ∅풇 = ퟐ(풇ퟐ − 풂ퟐ) 휶풂 + 휸풄 + ∅풇 = (풇ퟐ − 풂ퟐ) But 풇ퟐ − 풂ퟐ = ퟐ휸풄 휶풂 + 휸풄 + ∅풇 = ퟐ휸풄 ∴ 휶풂 + ∅풇 = 휸풄 [ퟏퟎ] 12 Brian Ekanyu From triangle BCD 푫푪̅̅̅̅ = 풃, 푴푵̅̅̅̅̅ = 휷 푩푪̅̅̅̅ = 풅, 푲푳̅̅̅̅ = 휹 푩푫̅̅̅̅̅ = 풇, 푬푭̅̅̅̅ = ∅ Considering triangle BDN 풃 푩푵̅̅̅̅̅ = 풉 , 푫푵̅̅̅̅̅ = ( − 휷), 푩푫̅̅̅̅̅ = 풇 ퟒ ퟐ 푩푵̅̅̅̅̅ퟐ + 푫푵̅̅̅̅̅ퟐ = 푩푫̅̅̅̅̅ퟐ 풃 ퟐ 풉 ퟐ + ( − 휷) = 풇ퟐ ퟒ ퟐ 풃ퟐ 풉 ퟐ = 풇ퟐ − + 휷풃 − 휷ퟐ [ퟏퟏ] ퟒ ퟒ Considering triangle BCN 풃 푩푵̅̅̅̅̅ = 풉 , 푪푵̅̅̅̅ = ( + 휷), 푩푪̅̅̅̅ = 풅 ퟒ ퟐ A New Theorem on Orthogonal Quadrilaterals 13 푩푵̅̅̅̅̅ퟐ + 푪푵̅̅̅̅ퟐ = 푩푪̅̅̅̅ퟐ 풃 ퟐ 풉 ퟐ + ( + 휷) = 풅ퟐ ퟒ ퟐ 풃ퟐ 풉 ퟐ = 풅ퟐ − − 휷풃 − 휷ퟐ [ퟏퟐ] ퟒ ퟒ Equating [ퟏퟏ] and [ퟏퟐ] 풃ퟐ 풃ퟐ 풇ퟐ − + 휷풃 − 휷ퟐ = 풅ퟐ − − 휷풃 − 휷ퟐ ퟒ ퟒ ퟐ휷풃 = 풅ퟐ − 풇ퟐ [ퟏퟑ] Considering triangle BDK 풅 푩푲̅̅̅̅̅ = ( − 휹), 푫푲̅̅̅̅̅ = 풉 , 푩푫̅̅̅̅̅ = 풇 ퟐ ퟓ 푩푲̅̅̅̅̅ퟐ + 푫푲̅̅̅̅̅ퟐ = 푩푫̅̅̅̅̅ퟐ 풅 ퟐ ( − 휹) + 풉 ퟐ = 풇ퟐ ퟐ ퟓ 풅ퟐ 풉 ퟐ = 풇ퟐ − + 휹풅 − 휹ퟐ [ퟏퟒ] ퟓ ퟒ Considering triangle CDK 풅 푪푲̅̅̅̅ = ( + 휹), 푫푲̅̅̅̅̅ = 풉 , 푪푫̅̅̅̅ = 풃 ퟐ ퟓ 푪푲̅̅̅̅ퟐ + 푫푲̅̅̅̅̅ퟐ = 푪푫̅̅̅̅ퟐ 풅 ퟐ ( + 휹) + 풉 ퟐ = 풃ퟐ ퟐ ퟓ 풅ퟐ 풉 ퟐ = 풃ퟐ − − 휹풅 − 휹ퟐ [ퟏퟓ] ퟓ ퟒ Equating [ퟏퟒ] and [ퟏퟓ] 풅ퟐ 풅ퟐ 풇ퟐ − + 휹풅 − 휹ퟐ = 풃ퟐ − − 휹풅 − 휹ퟐ ퟒ ퟒ ퟐ휹풅 = 풃ퟐ − 풇ퟐ [ퟏퟔ] Considering triangle BCE 풇 푪푬̅̅̅̅ = 풉 , 푩푬̅̅̅̅ = ( − ∅) , 푩푪̅̅̅̅ = 풅 ퟔ ퟐ 푪푬̅̅̅̅ퟐ + 푩푬̅̅̅̅ퟐ = 푩푪̅̅̅̅ퟐ 풇 ퟐ 풉 ퟐ + ( − ∅) = 풅ퟐ ퟔ ퟐ 풇ퟐ 풉 ퟐ = 풅ퟐ − + ∅풇 − ∅ퟐ [ퟏퟕ] ퟔ ퟒ 14 Brian Ekanyu Considering triangle CDE 풇 푪푬̅̅̅̅ = 풉 , 푫푬̅̅̅̅ = ( + ∅), 푪푫̅̅̅̅ = 풃 ퟔ ퟐ 푪푬̅̅̅̅ퟐ + 푫푬̅̅̅̅ퟐ = 푪푫̅̅̅̅ퟐ 풇 ퟐ 풉 ퟐ + ( + ∅) = 풃ퟐ ퟔ ퟐ 풇ퟐ 풉 ퟐ = 풃ퟐ − − ∅풇 − ∅ퟐ [ퟏퟖ] ퟔ ퟒ Equating [ퟏퟕ] and [ퟏퟖ] 풇ퟐ 풇ퟐ 풅ퟐ − + ∅풇 − ∅ퟐ = 풃ퟐ − − ∅풇 − ∅ퟐ ퟒ ퟒ ퟐ∅풇 = 풃ퟐ − 풅ퟐ [ퟏퟗ] Adding [ퟏퟑ], [ퟏퟔ] and [ퟏퟗ] ퟐ휷풃 + ퟐ휹풅 + ퟐ∅풇 = 풅ퟐ − 풇ퟐ + 풃ퟐ − 풇ퟐ + 풃ퟐ − 풅ퟐ ퟐ휷풃 + ퟐ휹풅 + ퟐ∅풇 = ퟐ(풃ퟐ − 풇ퟐ) 휷풃 + 휹풅 + ∅풇 = 풃ퟐ − 풇ퟐ But 풃ퟐ − 풇ퟐ = ퟐ휹풅 휷풃 + 휹풅 + ∅풇 = ퟐ휹풅 ∴ 휷풃 + ∅풇 = 휹풅 [ퟐퟎ] Subtracting [ퟏퟎ] from [ퟐퟎ] 휷풃 − 휶풂 = 휹풅 − 휸풄 ∴ 휶풂 + 휹풅 = 휷풃 + 휸풄 (Q.E.D) REFERENCES [1] Josefsson, Martin(2016), “Properties of Pythagorean quadrilaterals,” The Mathematical Gazette, 100( July) 213-224 [2] Weisstein, Eric W. (2010). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. Pp. 375-377. ISBN 9781420035223. [3] Bottomly, Henry. “Medians and Area Bisectors of a triangle. [4] Johnson, Roger A (2007)[1960], Advanced Euclidean Geometry, Dover, and ISBN 978-0-486-46237-0. [5] Eritrea’s theorem. The International Journal of Mathematics Research Vol.4, No. 4 2012 edition[pp. 455 − 461]. .
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