{"id":156082,"date":"2022-03-26T01:21:43","date_gmt":"2022-03-25T19:51:43","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/apollonius-theorem-statement-derivation-equation-examples-faqs\/"},"modified":"2023-05-08T14:25:54","modified_gmt":"2023-05-08T08:55:54","slug":"apollonius-theorem-statement-derivation-equation-examples-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/","title":{"rendered":"Apollonius Theorem &#8211; Statement, Derivation, Equation, Examples &#038; FAQs"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Apollonius_Theorem\" title=\"Apollonius Theorem\">Apollonius Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Statement_and_Proof_of_Apollonius_Theorem\" title=\"Statement and Proof of Apollonius&#8217; Theorem\">Statement and Proof of Apollonius&#8217; Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Apollonius_Theorem_Statement\" title=\"Apollonius\u2019 Theorem Statement\">Apollonius\u2019 Theorem Statement<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Apollonius_Theorem_Proof\" title=\"Apollonius\u2019 Theorem Proof\">Apollonius\u2019 Theorem Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Statement_and_Proof_by_the_Pythagorean_Theorem\" title=\"Statement and Proof by the Pythagorean Theorem\">Statement and Proof by the Pythagorean Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Statement_and_Proof_by_Vectors\" title=\"Statement and Proof by Vectors\">Statement and Proof by Vectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#Examples\" title=\"Examples\">Examples<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/apollonius-theorem\/#FAQs\" title=\"FAQs\">FAQs<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Apollonius_Theorem\"><\/span>Apollonius Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The theorem states that for every two circles there is a unique line that passes through their centers and is perpendicular to their common diameter. In mathematics, the Apollonius theorem is a statement in plane geometry that states that a circle is the locus of points equidistant from two other points. The theorem is attributed to the Greek mathematician Apollonius of Perga.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156081 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/apollonius-theorem-statement-derivation-equation-examples-faqs.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/apollonius-theorem-statement-derivation-equation-examples-faqs.jpg?v=1648237898 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/apollonius-theorem-statement-derivation-equation-examples-faqs-300x212.jpg?v=1648237898 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Statement_and_Proof_of_Apollonius_Theorem\"><\/span>Statement and Proof of Apollonius&#8217; Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Apollonius&#8217; theorem states that for every three points in a plane there exists a unique circle that passes through all three points.<\/p>\n<p><strong>Proof: <\/strong>We will use proof by contradiction. Suppose that there exists a circle that passes through all three points but that is not unique. This would mean that there are two circles that pass through all three points. But then the points would not be unique, which is a contradiction. Therefore, there must be a unique circle that passes through all three points.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Apollonius_Theorem_Statement\"><\/span>Apollonius\u2019 Theorem Statement<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let ABC be a right triangle with right angle C. If a and b are the lengths of the other two sides, then Apollonius\u2019 theorem states that<\/p>\n<p><strong>a2 + b2 = c2.<\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Apollonius_Theorem_Proof\"><\/span>Apollonius\u2019 Theorem Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let \\(A\\) and \\(B\\) be points on a circle and let \\(P\\) be the point on the circle that is the midpoint of \\(AB\\).<\/p>\n<p>Then, \\(AP = PB\\).<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Statement_and_Proof_by_the_Pythagorean_Theorem\"><\/span>Statement and Proof by the Pythagorean Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A right triangle has the length of its longest side, or hypotenuse, as its hypotenuse. The other two sides are the short side and the long side. The Pythagorean theorem states that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse. This theorem is represented by the equation: a^2 + b^2 = c^2.<\/p>\n<p>To prove the Pythagorean theorem, we can use a right triangle that has been constructed out of unit squares. The length of the short side is 1 unit, the length of the long side is 2 units, and the length of the hypotenuse is 3 units. We can see that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse. 1^2 + 2^2 = 3^2.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Statement_and_Proof_by_Vectors\"><\/span>Statement and Proof by Vectors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Statement: If vectors a and b are both nonzero, then the dot product a \u00b7 b is positive.<\/p>\n<p>Proof:<\/p>\n<p>We will show that the dot product a \u00b7 b is positive if and only if a and b are both nonzero.<\/p>\n<p>If a and b are both nonzero, then the dot product a \u00b7 b is positive.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Examples\"><\/span>Examples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Apollonius\u2019 theorem is a fundamental theorem in geometry that states the following: if the perpendicular bisectors of the sides of a triangle intersect at a point, then the angles of the triangle are equal. This theorem is named after the ancient Greek mathematician Apollonius of Perga, who first proved it in the third century B.C.<\/p>\n<p>Apollonius\u2019 theorem can be used to prove many other properties of triangles. For example, it can be used to prove that the circumcenter of a triangle is the point at which the perpendicular bisectors of its sides intersect. This point is also the center of the circumcircle, which is the circle that passes through all of the triangle\u2019s vertices.<\/p>\n<p>The theorem can also be used to prove that the incenter of a triangle is the point at which the angle bisectors of its angles intersect. The incenter is the center of the incircle, which is the circle inscribed inside the triangle and tangent to each of its sides.<\/p>\n<p>Apollonius\u2019 theorem can also be used to prove that the orthocenter of a triangle is the point at which its altitudes intersect. An altitude of a triangle is a line that passes through a vertex and is perpendicular to the opposite side. The orthocenter is the point at which the three altitudes of a triangle intersect.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"FAQs\"><\/span>FAQs<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p>Q: What is Apollonius Theorem?<\/p>\n<p>A: Apollonius Theorem is a geometric theorem that relates the length of a median of a triangle to the lengths of its other sides.<\/p>\n<p>Q: How can Apollonius Theorem be stated?<\/p>\n<p>A: Apollonius Theorem states that in a triangle, the sum of the squares of the lengths of the two smaller sides is equal to twice the square of the length of the median that bisects the third side.<\/p>\n<p>Q: How can Apollonius Theorem be proved?<\/p>\n<p>A: One way to prove Apollonius Theorem is to use the Law of Cosines. Suppose a triangle ABC has sides of length a, b, and c, and that the median to side c has length m. Then, we can use the Law of Cosines to express m in terms of a, b, and c. Then, we can square both sides of this equation and use the Law of Cosines again to simplify the expression. The resulting equation is Apollonius Theorem.<\/p>\n<p>Q: What are some applications of Apollonius Theorem?<\/p>\n<p>A: Apollonius Theorem is useful in geometry and trigonometry. It can be used to solve problems involving triangles, such as finding the length of a median or determining whether a triangle is acute, right, or obtuse.<\/p>\n<p>Q: How can Apollonius Theorem be used to find the length of a median?<\/p>\n<p>A: To use Apollonius Theorem to find the length of a median, we first identify the triangle&#8217;s sides and the median in question. Then, we plug the lengths of the two smaller sides and the median into Apollonius Theorem and solve for the length of the median.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Apollonius Theorem The theorem states that for every two circles there is a unique line that passes through their centers [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Apollonius Theorem - Statement, Derivation, Equation, Examples & FAQs","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Apollonius Theorem statement, derivation, equations and solved examples for better understanding. 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