Geometry
Five centers of Triangles
Questions

    Read Symbol defintion

  • Q01 | - Diagrams : Five centers of triangles
  • Q02 | - Definition : Five centers of triangles
  • Q03 | - Incenter of triangle
  • Q04 | - Escenter of triangle
  • Q05 | - Excenter of triangle
  • Q06 | - Gravity center of triangle
  • Q07 | - Orthocenter of triangle
  • Q08 | - Prove that ex-center, gravity center and orthocenter are colinear.
  • Q09 | - Draw Ex-center E and orthocenter O in large triangle.
    • Let AOH be the height and EM be bisector of side BC.
    • Prove that EM : AO = 1 : 2.
  • Q10 | - Questions
  • Q11 | - Geometric terms
Answers


Q01. Diagrams
  • Top left diagram
    • Use A, B, C express triangle sides and a, b, c express the sides.
    • Triagle area = (height)*(base)/2.
  • Top mid diagram : Orthocenter
    • Three heights of triangle are concurrent
  • Top right diagram : Circumcenter or ex-center
    • Three bisectors of sides of triangle are concurrent.
  • Bottom left diagram : Gravity center or centroid
    • Three medians are concurrent.
  • Bottom mid diagram : In-center
    • Three bisectors of interior triangles .
  • Bottom right diagram : Es-center
    • One bisector of interior angle and two bisector of exterior angles are concurrent.
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Q02. Defintion
  • In-center
    • Three bisectors of interior angles of a triangle are concurrent.
    • In-center to three sides has equal distance.
  • Ex-center or circumcenter
    • Three bisectors of the sides a triangle are concurrent.
    • Ex-center to three vertices has equal distance.
  • Es-center
    • One bisector of interior angle with 2 bisectors of exterior angles are concurrent.
    • Es-center to three sides has same distance.
    • One triangle can have three es-centers.
    • Join the es-centers to make a special triangle.
  • Gravity center or centroid
    • Three medians of a triangle are concurrent.
    • From centroid G to vertex A = 2*(AM)/3 where M is mid point on BC.
    • Medain is the line from vertex to the mid point of the opssite side.
  • Orthocenter
    • Three altitudes or heights of a triangle are concurrent.
    • If AH is height, we can use AC as diameter to draw a circle passing H.
    • If AH is height, we can use AB as diameter to draw a circle passing H.
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Q03. In-center of triangle
  • Definition :
    • Three bisectors of interior angles meet at one point.
    • The center is inside the triangle.
  • Properties :
    • Incenter has same distance from the sides of triangle.
    • An incircle can be drawn to touch the sides of triangle.
  • Prove that three bisectors interior angles meet one point.
    • Construction : Draw diagram on paper
      • Draw bisector at angle A and angle B. These two bisectors meet at I.
      • From center I to side AB is D and to AC is E. Hence ID = IE.
      • From center I to side BC is F and to BA is D. Hence ID = IF.
      • Join I and C. Prove that IC is a bisector of angle C.
    • Proof
      • IE = IF. Angle IEC = angle IFC = 90 degrees
      • IC is common side of triangle IEC and IFC.
      • Hence triangle IEC is congruent to triangle IFC.
      • Hence angle ICF = angle ICE.
      • Hence line IC is also a bisector of angle C.
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Q04. Es-center of triangle
  • Definition :
    • Intersection of 2 bisectors of exterior angles and 1 bisector of interior angle.
    • The center is outside of the triangle.
    • A trinagle can have 3 escenters.
  • Properties :
    • Escenter has same distance from the sides of triangle.
    • An escircle can be drawn to touch thhree sides of triangle.
  • Prove that One bisector of interior angle and two bisectors of exterior angles are concurent.
    • Construction.
      • Draw bisector at exterior angle A and angle B. These two bisectors meet at P.
      • From center P to side CA is D and to AB is E. Hence PD = PE.
      • From center P to side CB is F and to BA is E. Hence PD = PF.
      • Join P and C. Prove that PC is a bisector of angle C.
    • Proof
      • PE = PF. Angle PDC = angle PFC = 90 degrees
      • PC is common side of triangle PDC and PFC.
      • Hence triangle PDC is congruent to triangle PFC.
      • Hence angle PCD = angle PCF.
      • Hence line PC is also a bisector of angle C.
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Q05. Ex-center of triangle
  • Definition :
    • Three bisectors of three sides meet at one point.
    • The center is insdie the triangle It is also called circumcenter.
  • Properties :
    • Excenter has same distance from the vertices of triangle.
    • An excircle can be drawn to pass the vertices of triangle.
  • Prove that three bisectors of three sides meet one point.
    • Construction
      • Draw bisector of side AB at D and bisector of side BC at E.
      • These two bisectors meet at Q.
      • From center Q to vertex A and vertex B are equal. Hence QA = QB.
      • From center Q to vertex B and vertex C are equal. Hence QB = QC.
      • Draw line from Q to mid point F of AC.
      • Prove that QF is a bisector of side AC.
    • Proof
      • QA = QC. AF = FC.
      • QF is common side of triangle QAF and QCF.
      • Hence triangle QAF is congruent to triangle QCF.
      • Hence angle QFA = angle QFC = 90 degrees since AC is straight line.
      • Hence line QF is a bisector of side AC.
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Q06. Gravity center of triangle
  • Definition :
    • Three medians of a triangle meet at one point G. It is also called centroid.
    • If AD is one median. then D is mid point of BC.
    • Then we have DG = GA/2 = AD/3.
  • Properties
  • Prove that three medians meet at one pint.
    • Construction
      • Draw triangle ABC.
      • Draw CF as one median and BE as another median.
      • CF and BE meet at G.
      • Draw line AG and extend to D on side BC.
      • If D is mid-point, then AD is a median and passes point G.
      • Proof is complete. How to prove D is mid-point of BC ?
    • Proof
      • Extend line AD to H so that AG = GH.
      • In triangle ABH, FG = BH/2 and FG parallel to BH. GC parallel to BH.
      • In triangle ACH, EG = CH/2 and FG parallel to CH. BG parallel to CH.
      • Hence BGCH is a parallelogram.
      • Bisect of the diagonal giving that BD = DC and DG = DH.
      • Hence D is mid-point of BC.
      • Hence AD is a median and passes G.
  • Prove that gravity center to vertex is (2*median)/3.
    • From above, we see that GD = DH = AG/2. Hence AG = 2*DG.
    • AD = AG + GD = 3*GD. Hence GD = AD/3.
    • Hence AG = 2*AD/3. AD is a median.
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Q07. Orthocenter of triangle
  • Definition :
    • Three heights of a triangle meet at one point O.
    • If AD is one height. then angle ADB = 90.
    • Use AB as a diameter making a circle which will pass point D.
    • Use AC as a diameter making a circle which will pass point D.
  • Properties : Constructions
    • Draw line through A parallel to BC.
    • Draw line through B parallel to AC.
    • Draw line through C parallel to AB.
    • Let PQ parallel to AC. Let PR parallel to BC. Let QR parallel to AB.
    • It forms a triangle PQR whose excenter is the orthocenter of triangle ABC.
  • Prove that three medians meet at one pint.
    • Construction
      • As above, draw triangle ABC and triangle PQR.
      • Draw CF perpendicular to AB.
      • Draw BE perpendicular to AC.
      • CF and BE meet at O which is the orthocenter.
      • Draw line AO and extend to D on side BC,
      • IF AD is perpendicular to BC. The proof is done.
    • Proof : AD is perpendiuclar to BC.
      • Since O is ex-center of triangle PQR, hence OA is perpendicular to PR.
      • PR is parallel to BC by construction.
      • Hence AD is perpendicular to BC.
      • AD passes to O and AD is a height of triangle ABC.
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Q08. Prove that ex-center, gravity center and orthocenter are colinear.
Construction method
  • Draw a very large triangle ABC.
  • Draw the ex-center E of triangle ABC.
  • Draw the gravity center G of triangle ABC.
  • Draw the ortho-center O of triangle ABC.
  • We can see that points E, G, O are on a straight line.
Coordinate geometric method
  • Draw a triangle ABC using (x1,y1), (x2,y2), (x3,y3).
  • Find coordinate of the ex-center E(xe,ye) of triangle ABC.
  • Find coordinate of the gravity center G(xg,yg) of triangle ABC.
  • Find coordinate of the ortho-center O(xO,yO) of triangle ABC.
  • Find s1 = slope of EG and s2 = slope EO.
  • If s1 = s2, then the proof is complete
Geometrical method
  • Construction
    • Draw ex-center E, gravity center G and orthocenter O.
    • Let AM is redian and AH is the height.
    • Draw mid point P on AG.
    • Draw mid point Q on OG.
    • Prove that triangle PQG is congruent to GEM.
  • Proof
    • PQ is parallel to AO or AH.
    • Angle EMG = angle GPQ.
    • PG = AG/2 = GM
    • PQ = AO/2 = EM (See Q09)
    • Hence triangle PQG is congruent to EGM.
    • Hence angle EGM = angle PGQ.
    • Since AGM is a line, hence EGO is a line
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Q09. Ex-center E and orthocenter O. AOH is height. EM bisector of BC. Prove that AO = 2*EM
  • Construction
    • Draw large triangle. Draw ex-center E and orthocenter O.
    • Draw AH perpendicur BC and EM bisect BC.
    • Prove that AO = 2*EM.
  • Proof
    • Let N be mid point on AB.
    • Hence MN parallel to AC and MN = AC/2
    • Draw mid point P on AO and Q on OC.
    • Hence PQ parallel to AC and PQ = BC/2.
    • Since PQ parallel to MN and PQ = MN.
    • Also EM parallel to AO and EN parallel to OQ.
    • Hence triangle EMN is congruent to POQ.
    • Hence OP = EM = AO/2
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Q10. Questions
  • 1. How to draw bisector of an angle ?
  • 2. How to draw bisector of a line ?
  • 3. What is median of a triangle ?
  • 4. What is altitude of a triangle ?
  • 5. What is the property of bisector of an angle ?
  • 6. What is the property of bisector of a line ?
  • 7. A point P to two fixed points A and B have same distance. What is locus P ?
  • 8. A point P to two arms of an angle keeps same distance. What is locus of P ?
  • 9. Prove that three bisectors of interior angles of triangle meet one point.
  • 10 Prove that three bisectors of three sides of triangle meet one point.
  • 11 Draw gravity center G of triangle.
    • Let AGM be one median of the triangle.
    • Prove that MG : GA : MA = 1 : 2 : 3.
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Q11. Geometric terms about triangle
  • 1. Colinear of points : Three or more points on same line.
  • 2. Concurrent of three lines : Three lines meet at one point
  • 3. Congruent of triangle : Two triangles are idential sides (SSS)
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