What you’ll learn
Construct angle bisectors.
Key idea
Use the same second opening at both hits, sufficiently large for their arcs to cross inside the angle.
Both equal-distance pairs and common ray are needed; the second opening must permit an inside crossing.
Worked example
Enter the exact value or ordered values in the format requested below.
Angle ABC has vertex B=(0,0); rays BA/BC pass through D=(5,2), E=(5,-2). One B-centered arc hits D/E, and equal inner arcs meet at F inside. Construct the bisector. Enter first_center=...; ray_hits=...; second_centers=...; second_radius=...; ray=... . State second-radius condition relative to DE.
- BD squared = BE squared = 29; DE=4, so choosing equal second opening 4>2 produces an inside crossing.
- BD=BE, DF=EF and BF=BF give SSS of BDF/BEF, hence BF divides the angle equally.
Answer: first_center=B; ray_hits=D,E; second_centers=D,E; second_radius=>DE/2; ray=BF
Quick Check
Choose which candidate representation satisfies the requirements of 'Construct angle bisectors' and justify the choice.
Angle ABC has vertex B=(0,0); rays BA/BC pass through D=(5,2), E=(5,-2). One B-centered arc hits D/E, and equal inner arcs meet at F inside. Plan 1 keeps both opening pairs equal and draws BF. Plan 2 draws DE from arbitrary ray points. Plan 3 uses unequal openings at D/E before BF. Choose and justify by congruence.