Given a sentence in Tagalog language, translate the sentence to English language keeping the meaning of the original sentence intact.

Q: Tagalog sentence: Ang point M ay ang midpoint ng Arc BC sa circle c. Kaya, ang dalawang Chords BM at CM ng c bilog ay may pantay na haba. Kaya, ang chordal angles ng mga dalawang Chords ay dapat pantay-pantay, sa ibang salita, mayroon kaming &lt; BAM = &lt; CAM. Kaya, ang point M kasinungalingan sa anggulo panggitnang guhit ng anggulo BAC.

A: The point M is the midpoint of the arc BC on the circle c. Thus, the two chords BM and CM of the circle c have equal length. Hence, the chordal angles of these two chords must also be equal; in other words, we have &lt; BAM = &lt; CAM. Hence, the point M lies on the angle bisector of the angle BAC. Similarly, the point N lies on the angle bisector of the angle BAD. But, of course, the angle bisector of the angle BAD is the exterior angle bisector of the angle BAC; hence, since the interior and the exterior angle bisector of an angle are always perpendicular to each other, the angle bisector of the angle BAC and the angle bisector of the angle BAD are perpendicular to each other. Hence, &lt; MAN = 90°.
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Q: Tagalog sentence: Ang ibang tao ay maaaring magustuhan ang aspektong sosyal ng paglalaro ng blackjack sa isang mesa, kasama ng mga kaibigan, samantalang ang iba ay maaaring piliing mag-relax sa paglalaro ng slot machine o VLT. Magpatuloy sa pagbasa para malaman kung ano ang kahulugan ng pagsusugal para sa iba’t ibang tao.

A: Some people might like the social aspect of playing blackjack at a table with friends, while others might prefer to relax by themselves at a slot machine or VLT. Read on to explore what gambling can mean for different people.
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Q: Tagalog sentence: ... isang mahaba at matagal imbestigasyon, na kung saan kinakailangan marami kasakdalan ng detalye pati na rin ang katalinuhan ng mga kuru-kuro.

A:
... a long and sustained investigation, which required much perfection of detail as well as brilliance of conception.
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